A test for stationarity for spatio-temporal

data

Soutir Bandyopadhyay∗, Carsten Jentsch†and Suhasini Subba Rao‡

January 27, 2016

Abstract

Many random phenomena in the environmental and geophysical sciences are func-

tions of both space and time; these are usually called spatio-temporal processes. Typ-

ically, the spatio-temporal process is observed over discrete equidistant time and at

irregularly spaced locations in space. One important aim is to develop statistical mod-

els based on what is observed. While doing so a commonly used assumption is that

the underlying spatial-temporal process is stationary. If this assumption does not hold,

then either the mean or the covariance function is misspecified. This can, for example,

lead to inaccurate predictions. In this paper we propose a test for spatio-temporal

stationarity. The test is based on the dichotomy that Fourier transforms of stochastic

processes are near uncorrelated if the process is second order stationarity but corre-

lated if the process is second order nonstationary. Using this as motivation, a Discrete

Fourier transform for spatio-temporal data over discrete equidistant times but on irreg-

ularly spaced spatial locations is defined. Two statistics which measure the degree of

correlation in the Discrete Fourier transforms are proposed. These statistics are used

to test for spatio-temporal stationarity. It is shown that the same statistics can also be

adapted to test for the one-way stationarity (either spatial or temporal stationarity).

The proposed methodology is illustrated with a small simulation study.

Key words and phrases: Fourier Transforms; Irregular sampling; Nonstationar-

ity; Stationary random fields; Spectral density; Orthogonal samples.

∗Department of Mathematics, Lehigh University, Bethlehem, PA, U.S.A. sob210@lehigh.edu†Department of Economics, University of Mannheim, Mannheim, Ger-

many.cjentsch@staff.mail.uni-mannheim.de‡Department of Statistics, Texas A&M University, College Station, TX, U.S.A. suhasini@stat.tamu.edu

1

1 Introduction

Several environmental and geophysical phenomena, such as tropospheric ozone and precip-

itation levels, are random quantities depending on both space and time. Since, in practice,

it is only possible to observe the process on a finite number of locations in space, sjnj=1

and typically over discrete equidistant time t = 1, . . . , T , one aim in the geosciences is

to develop statistical models based on what is observed. Typically this is done by fit-

ting a parametric space-time covariance function defined on Zt(s); s ∈ Rd, t ∈ Z to the

data. Such models can, then, be used for prediction and forecasting at unobserved loca-

tions; see Gneiting et al. [2006] and Sherman [2010] for an extensive survey on space-time

models. In this context, an assumption that is often used is that the underlying spatio-

temporal process Zt(s); s ∈ Rd, t ∈ Z is stationary, in the sense that E[Zt(s)] = µ and

cov[Zt(s1), Zτ (s2)] = κτ−t(s2 − s1). If this assumption does not hold, then either the mean

or the covariance function is misspecified which, for example, can lead to inaccurate pre-

dictions. Therefore, in order to understand the underlying structure of the spatio-temporal

process correctly we should test for second order stationarity of the spatio-temporal process

first. Furthermore, given that often the size of the data sets are extremely large, the test

should be computationally feasible. The aim of this paper is to address these issues.

Before we describe the proposed procedure, we start by surveying some of the tests for

stationarity that exist in the literature. One of the earliest tests for temporal stationarity is

given in Priestley and Subba Rao [1969]. More recently, several tests for temporal station-

arity have been proposed; these include von Sachs and Neumann [1999], Paparoditis [2009],

Paparoditis [2010], Dette et al. [2011], Dwivedi and Subba Rao [2011], Jentsch [2012], Nason

[2013], Lei et al. [2015], Jentsch and Subba Rao [2015], Cho [2014] and Puchstein and Preuss

[2016].

For spatial data, Fuentes [2006] generalizes the test proposed in Priestley and Subba Rao

[1969] to spatial data defined on a grid and Epharty et al. [2001] proposes a test for spatio-

temporal stationarity for data defined on a spatio-temporal grid. However, if the spatial data

is defined on irregular locations (typically, a more realistic scenario), then there exists only

a few number of tests. As far as we are aware, the first test for spatio-temporal stationarity,

where the spatial component of the data is observed at irregular locations is proposed in Jun

and Genton [2012]. More recently, Bandyopadhyay and Subba Rao [2016] propose a test for

spatial stationarity where the data is observed at irregular locations.

In this paper we develop a test for spatio-temporal stationarity, where time is defined

on Z and the locations are irregular on Rd. Our procedure is heavily motivated by the

tests in Epharty et al. [2001], Dwivedi and Subba Rao [2011], Jentsch and Subba Rao [2015]

and Bandyopadhyay and Subba Rao [2016], which use a Fourier transform of the data to

discriminate between the stationary and nonstationary behavior. To motivate our approach

2

let us consider the Cramer representation of a stationary stochastic process, which states

that a second order stationary stochastic process, Zt(s), s ∈ Rd, t ∈ Z can always be

represented as

Zt(s) =1

2π

∫ 2π

0

∫Rd

exp(itω) exp(is′Ω)dZ(Ω, ω), (1)

where, Z(Ω, ω) is a stochastic process with orthogonal increments, i.e., E[dZ(Ω1, ω2)dZ(Ω3, ω4)] =

0 if Ω1 6= Ω3 or ω2 6= ω4 and E[|dZ(Ω1, ω2)|2] = dF (Ω, ω) = f(Ω, ω)dΩdω, where f denotes

the spectral density (and the second equality only holds if the derivative of F exists); see

Subba Rao and Terdik [2016]. On the other hand, if the increments are correlated, then

the process is not second order stationary (see for example, Gladyshev [1963], Goodman

[1965], Yaglom [1987] Lii and Rosenblatt [2002], Hindberg and Olhede [2010], Gorrostieta

et al. [2016]). Furthermore, the increment process yields information about the station-

arity of the process in particular domains. For example, suppose the process is spatially

stationary (but not necessarily temporally stationary), then E[dZ(Ω1, ω2)dZ(Ω3, ω2)] = 0 if

Ω1 6= Ω3. Conversely, if the process is temporally stationary but not spatially stationary,

then E[dZ(Ω1, ω2)dZ(Ω1, ω4)] = 0 if ω2 6= ω4.

Of course in practice the increment process is unobserved. However, in time series analysis

the Discrete Fourier transform (DFT) of a time series is considered as an estimator of the

increments in the increment process and shares many of its properties. In particular, the

Discrete Fourier transform of a stationary time series is a ‘near uncorrelated’ transformation,

thus mirroring the properties of the increment process. In Dwivedi and Subba Rao [2011] and

Jentsch and Subba Rao [2015] we use the Discrete Fourier transform to test for stationarity.

On the other hand, the Fourier transform for spatial data defined on irregular locations

is not uniquely defined. However, Matsuda and Yajima [2009] and Bandyopadhyay and

Lahiri [2009] define a Fourier transform on spatial data with irregular locations which can be

shown to share similar properties as the increment process when the locations are uniformly

distributed. In Bandyopadhyay and Subba Rao [2016] we exploit this property to test for

spatial stationarity. In this paper we combine both these transformations to define a Discrete

Fourier transform for spatio-temporal data that is defined over discrete time but on irregular

spatial locations. We show that this space-time Discrete Fourier transform satisfies many of

the properties of (1); in particular under stationarity the space-time DFT is asymptotically

uncorrelated, whereas under nonstationarity this property does not hold. In this paper we

use this dichotomy to define tests for stationarity for spatio-temporal processes.

In Section 2.1 we review the test for temporal stationarity proposed in Dwivedi and

Subba Rao [2011] and Jentsch and Subba Rao [2015]. In Section 2.2 we review the test

for spatial stationarity proposed in Bandyopadhyay and Subba Rao [2016]. We note that

3

there are some fundamental differences between the testing methodology over time compared

to the testing methodology over space. The first is that over discrete time the Fourier

transform can only be defined over a compact support, whereas the Fourier transform on

space can be defined over Rd (see the range of the integrals in (1)). This leads to significant

differences in the way that the test statistics can be defined. Furthermore, both the test over

time and the test over space involve variances which need to be estimated. In the test for

stationarity proposed in Jentsch and Subba Rao [2015] we used the stationary bootstrap to

estimate the variance, however using a block-type bootstrap for the spatial stationarity test

was computationally too intensive. Instead we used the method of orthogonal samples to

estimate the variance, which led us to a computationally feasible test statistic. This method

can also be applied to testing for temporal stationarity, and in Section 2.3 we use the same

method to estimate the variance in the test for temporal stationarity. The results in this

section motivate our proposed procedure in the later sections.

In Section 3 we turn to the spatio-temporal data. We define a Fourier transform (to reduce

notation we call it a “DFT”), which is over irregular locations in space, but for equidistant

discrete time. We obtain the correlation properties of the DFTs in the case of (i) spatial and

temporal stationarity, (ii) spatial stationarity (but not necessarily temporally stationary),

(iii) temporal stationarity (but not necessarily spatially stationary) and (iv) both temporal

and spatial nonstationarity. We show that each case has its own specific characterization in

terms of the DFTs. In Section 4 we use the differing behaviors to construct the test statistics.

Similar to both the stationarity test over space and the stationarity test over time, the test

here involves unknown variances, which are estimated using orthogonal samples. This means

the test statistic can be calculated in O(n2T log T ) computing operations. In Section 5 we

apply the methodology for testing one-way stationarity (stationary in one domain but not

necessarily stationary on the other domain). Some results relating to the test statistics are

in Section 6. Our proposed methods are illustrated with simulations in Section 7. A rough

outline of the proofs are given in the supplementary material.

2 Using the DFT to test for stationarity over time or

space

Our test for spatio-temporal stationarity is based on some of the ideas used to develop the

temporal and spatial tests in Dwivedi and Subba Rao [2011], Jentsch and Subba Rao [2015]

and Bandyopadhyay and Subba Rao [2016]. Therefore in Sections 2.1 and 2.2 we review some

pertinent features of these tests. In Section 2.3 we apply some of the methods discussed in

Section 2.2 to test for temporal stationarity.

4

2.1 Testing for temporal stationarity

Let us suppose that Xt is a stationary time series where ch = cov[Xt, Xt+h] and∑

h |hch| <∞. Given that we observe XtTt=1, we define the DFT of a time series XtTt=1 as JT (ωk) =

1√2πT

∑Tt=1 Xte

itωk , where ωk = 2πk/T are the so called Fourier frequencies. Dwivedi and

Subba Rao [2011] and Jentsch and Subba Rao [2015] exploit the property that under sta-

tionarity (and the short memory condition stated above) the DFT JT (ωk) is a ‘near un-

correlated sequence’ whose variance is approximately equal to the spectral function

f(ωk) =1

2π

∑h∈Z

ch exp(−ihωk).

We now briefly describe the procedure proposed in Jentsch and Subba Rao [2015] to test

for stationarity of a multivariate time series. To understand the pertinent features of the

test, we focus on the univariate case. Jentsch and Subba Rao [2015] estimates the spectral

density from the data (we denote the estimator, by smoothing the periodogram, as fT ), use

this to ‘standardize’ the DFT and define the estimator of the covariance between the DFTs

at ‘lag’ r as

CT (r, `) =1

T

T∑k=1

exp(i`ωk)JT (ωk)JT (ωk+r)√fT (ωk)fT (ωk+r)

. (2)

If we set ` = 0, then CT (r, 0)r can be viewed as the sample ‘autocovariance’ of the sequence

JT (ωk)/fT (ωk)1/2Tk=1 over frequency. CT (r, 0) was used as the basis of the test statistic.

Jentsch and Subba Rao [2015] showed that the approximate ‘variance’ (in terms of the

limiting distribution) of <CT (r, `) and =CT (r, `) (where <x and =x denote the real and

imaginary parts of x) is v`(ωr), where

v`(ωr) =1

2[1 + δ`,0 + κ`(ωr)]

with

κ`(ωr) =1

2π

∫ 2π

0

∫ 2π

0

f4(λ1 + ωr, λ2,−λ2 − ωr)√f(λ1)f(λ1 + ωr)f(λ2)f(λ2 + ωr)

exp[i`(λ1 − λ2)]dλ1dλ2 (3)

and f4 is the fourth order spectral density, which is defined as

f4(ω1, ω2, ω3) =1

(2π)3

∑h1,h2,h3∈Z

κh1,h2,h3 exp(−ih1ω1 − ih2ω2 − ih3ω3),

5

where κh1,h2,h3 = cum[X0, Xh1 , Xh2 , Xh3 ]. Moreover, for fixed ` and m and under suitable

mixing conditions we have

√T[<CT (1, `),=CT (1, `), . . . ,<CT (m, `),=CT (m, `)

]D→ N (0, v`(0)I2m) , (4)

as T →∞, where I2m denotes the identity matrix of order 2m. Note that v`(ωr)→ v`(0) as

T → ∞. We observe that in the case where the time series is stationary and Gaussian, we

have κ`(0) = 0 and v`(0) = 12(1+δ0,`); consequently, CT (r, ` is pivotal (does not depend on

any nuisance parameters). In contrast, if the time series is stationary but non-Gaussian, the

term κ`(0) does not vanish (indeed assuming Gaussianity when the process is not Gaussian

can lead to inflated type I errors in the test statistic defined below); compare Section 6.2 in

Jentsch and Subba Rao [2015].

Based on (4) for a fixed `, the test statistic is then defined as

Tm = Tm∑r=1

|CT (r, `)|2

v`(0), (5)

which under the null of stationarity, asymptotically has a chi-squared distribution with 2m

degrees of freedom. In practice v`(0) is unknown. Therefore, Jentsch and Subba Rao [2015]

uses the stationary bootstrap, proposed in Politis and Romano [1994], to estimate v`(0)

(actually they estimate v`(ωr)). In Section 2.3 of this paper we describe an alternative method

for estimating v`(0), which is a computationally much faster method. Note that in Jentsch

and Subba Rao [2015] a more general statistic based on CT (r, `); r = 1, . . . ,m, ` = 1, . . . , Lis proposed.

To understand how CT (r, `) behaves in the case the process is nonstationary we assume

that the time series ‘evolves’ slowly over time (a notion that was first introduced in Priestley

[1965]). To obtain the asymptotic limit of CT (r, `) we use the rescaling device introduced

in Dahlhaus [1997], where it was used to develop and study the class of locally stationary

time series. More precisely, we consider the class of locally stationary processes Xt,T,whose covariance structure changes slowly over time such that there exist smooth functions

κr;·r which can approximate the time-varying covariance, i.e., |cov(Xt,T , Xt+h,T )−κh; tT| ≤

T−1ρh, where ρh is such that∑

h |hρh| < ∞ (see Dahlhaus [2012]). Further, we define

the time-dependent spectral density Fu(ω) = 12π

∑h∈Z κh;ue

−ihω. Under this set-up we have

CT (r, `)P→ A(r, `), where

A(r, `) =1

2π

∫ 2π

0

∫ 1

0

Fu(ω)

f(ω)exp(−i2πru) exp(i`ω)dudω (6)

and f(ω) =∫ 1

0Fu(ω)du.

6

2.2 Testing for spatial stationarity

In Bandyopadhyay and Subba Rao [2016] our objective is to test for spatial stationarity

for a spatial random process Z(s); s ∈ Rd, observed only at a finite number of irreg-

ularly spaced locations, denoted as sjnj=1, in the region [−λ/2, λ/2]d, i.e., we observe

(sj, Z(sj)); j = 1, . . . , n. Suppose Z(s); s ∈ Rd is spatially stationary and denote

c(v) = cov[Z(s), Z(s + v)]. Analogous to the stationarity test for a time series described in

Section 2.1 we test for spatial stationarity by checking for uncorrelatedness of the Fourier

transforms. We note that the DFT of a discrete time time series (as described above) is a

linear one-to-one transformation between the time series in the time domain to the frequency

domain that can be easily inverted using the inverse DFT. On the other hand, when the loca-

tions are irregularly spaced, i.e. they are not on an equidistant grid on [−λ/2, λ/2]d, there is

no unique way to define the Fourier transform. Instead to test for stationarity, we use a suit-

able Fourier transform for irregularly sampled data which retains the near uncorrelated prop-

erty. More precisely, we define the Fourier transform as Jn(Ω) = λd/2

n

∑nj=1 Z(sj) exp(is′jΩ)

where Ω ∈ Rd (this Fourier transform was first defined in Matsuda and Yajima [2009] and

Bandyopadhyay and Lahiri [2009]). Note that the factor λd/2

nensures that the variance of

Jn(Ω) is non-degenerate when we let λ → ∞. Contrary to the time series case, we use Ω

instead of ω for spatial frequencies as we make use of both notations later for spatio-temporal

processes in Section 3.

Under the condition that the locations sj are independent and uniformly distributed

random variables on [−λ/2, λ/2]d and Z(s); s ∈ Rd is a fourth order stationary pro-

cess (with suitable short memory conditions), Bandyopadhyay and Subba Rao [2016] shows

that the Fourier transform at the ordinates Ωk = 2π(k1λ, . . . , kd

λ)′, k = (k1, . . . , kd) ∈ Zd,

i.e., Jn(Ωk)’s are ‘near uncorrelated’ random variables. For their variances, we have

var[Jn(Ωk)] = f(Ωk) +O( 1λ

+ λd

n), where

f(Ω) =

∫Rdc(s) exp(−is′Ω)dΩ

is the spectral density function of the spatial process. So far the results are very similar to

those in time series, however, because the spatial process is defined over Rd and not over

Zd, the spectral density f(Ω) is defined over Rd. For the same reason, f is no longer an

infinite sum, but becomes an integral. Furthermore, |f(Ω)| → 0 as ‖Ω‖2 →∞, where ‖ · ‖2

denotes the Euclidean norm (since the spatial covariance decays to zero sufficiently fast,

c(·) ∈ L2(Rd) and thus by Parseval’s inequality f ∈ L2(Rd)). Therefore, 1/√f(Ω) is not a

well defined function for all Ω ∈ Rd and unlike the discrete time series case, the standardized

Fourier transform Jn(Ωk)/√f(Ωk) is not a well defined quantity at all frequencies. Instead,

to measure the degree of correlation between DFTs, we have to avoid standardization and

7

we define the weighted covariance between the (non-standardized) Fourier transforms as

Aλ(g; r) =1

λd

a∑k1,...,kd=−a

g(Ωk)Jn(Ωk)Jn(Ωk+r)

−[

1

n2

a∑k1,··· ,kd=−a

g(Ωk)n∑j=1

Z2(sj) exp(−is′jΩr)

], (7)

where, r 6= 0, r = (r1, . . . , rd)′ ∈ Zd (with k and k + r defined analogously), g is a

given Lipschitz continuous function with supΩ∈Rd |g(Ω)| < ∞ and a satisfies (λa)d/n2 →0. In order to avoid the so called ‘nugget effect’ where the observations are corrupted by

measurement error (typically independent noise) we have subtracted the variance-type term

in the definition of (7).

We give some examples of functions g below.

Remark 2.1 Examples of g used in Bandyopadhyay and Subba Rao [2016] are functions of

the form g(Ω) = eiv′Ω, which is geared towards detecting changes in the spatial covariance at

lag v. However, unlike the case of regularly spaced locations, where we can detect changes at

integer lags, it is unclear which lags to use. For this reason in Bandyopadhyay and Subba Rao

[2016] we choose g(·) such that it can detect the aggregate change over L lags, namely g(Ω) =∑Lj=1 exp(iv′jΩ) (where vj is some grid within the main support of the covariance). We

should note that g(·) is similar to the weight function ei`ω[fT (ω)fT (ω + ωr)]−1/2 used in the

definition of CT (r, `) in (2).

We derive the sampling results under the mixed asymptotic framework, where λ → ∞ and

λd/n → 0, i.e., as the spatial domain grows, the number of observations should become

denser on the spatial domain (see, Hall and Patil [1994], Lahiri [2003], Matsuda and Yajima

[2009], Bandyopadhyay and Lahiri [2009], and Bandyopadhyay et al. [2015]). Under this

mixed asymptotic framework, we show in Theorem 3.1 of Bandyopadhyay and Subba Rao

[2016], that

E[Aλ(g; r)

]=

O(

1λd−b

∏d−bj=1 (log λ+ log |mj|)

), r ∈ Zd/0

1(2π)d

∫Ω∈Rd f(Ω)g(Ω)dΩ +O

(log λλ

+ 1n

), r = 0

where ad = O(n), a/λ→∞ as n→∞ and λ→∞, b = b(r) are the number of zero values

in the vector r, mj are the non-zero values in the vector r. From Section 3 (Theorem 3.3

treats the Gaussian case and the non-Gaussian case can be found at the bottom of Section

3), Bandyopadhyay and Subba Rao [2016], we have

c−1/2a,λ λd/2

[<Aλ(g; r1),=Aλ(g; r1), . . . ,<Aλ(g; rm),=Aλ(g; rm)

]′ D→ N (0, I2m) (8)

8

as `λ,a,n := log2 a(

log a+log λλ

)+ λd

n+ adλd

n2 + log3 λλ→ 0, where

ca,λ =1

2(2π)d

∫Df 2(Ω)

(|g(Ω)|2 + g(Ω)g(−Ω)

)dΩ

+1

2(2π)2d

∫D2

f4(Ω1,Ω2,−Ω2)g(Ω1)g(Ω2)dΩ1dΩ2,

D = [−2πa/λ, 2πa/λ]d and

f4(Ω1,Ω2,Ω3) =

∫R3d

κ(s1, s2, s3)e−i(s′1Ω1+s′2Ω2+s′3Ω3)ds1ds2ds2

is the (spatial) tri-spectral density and κ(s1, s2, s3) = cum[Z(0), Z(s1), Z(s2), Z(s3)] is the

fourth order cumulant analogous to κh1,h2,h3 in the time series case. We observe that unlike

CT (r, `) (defined in (2)), even in the case that the random field is stationary and Gaussian,

Aλ(g; r) is not asymptotically pivotal. This is because, unlike CT (r, `), in the definition of

Aλ(g; r) we could not standardize the Fourier transform Jn(Ω) such that f(Ω) crops up in

the asymptotics here. Therefore, even for Gaussian random fields, the variance ca,λ needs

to be estimated and if the random field is non-Gaussian then ca,λ additionally contains a

function of the fourth order spectral density.

In the following remark, we present an approach based on orthogonal samples, as proposed

by Subba Rao [2015b], to estimate complicated variances.

Remark 2.2 (Using orthogonal samples for variance estimation) The expression for

the variance Aλ(g; 0) given in the examples above, is rather unwieldy and difficult to estimate

directly. For example, in the case that the random field is Gaussian, one can estimate ca,λ by

replacing the integral with the sum∑a

k=−a and the spectral density function with periodogram

|Jn(Ωk)|2 (see Bandyopadhyay et al. [2015], Lemma 7.5). However, in the case that the pro-

cess is non-Gaussian this is not possible. Here we describe the method of orthogonal samples,

which can be used for both spatial and/or temporal data and it is a simple consistent method

for estimating the variance.

Let us suppose that AD(X) is an estimator of A where E[AD(X)]→ A and var[√DAD(X)] =

ν (where D = D(T, λ) is an appropriate scaling factor such that var[√DAD(X)] = O(1).

For some set B, the sample √DAD(X; j); j ∈ B (which is not necessarily real-valued) is the

orthogonal sample associated with AD(X) if (i) √DAD(X; j); j ∈ B and AD(X) are almost

uncorrelated but (ii) √DAD(X; j); j ∈ B has mean almost zero and var[

√DAD(X; r)] =

ν + o(1). Based on this we can estimate ν using

ν = σ2(√DAD(X; j); j ∈ B) =

D

2|B|∑j∈B

[(<AD(X; j)− A)2 + (=AD(X; j)− A)2

], (9)

9

and A = 12|B|∑

j∈B[<AD(X; j) + =AD(X; j)], where |B| denotes the cardinality of the set B.

Furthermore, if√D[AD(X)− A, <AD(X; j),=AD(X; j); j ∈ B] D→ N(0, νI2|B|+1), then

√D

[AD(X)− A]√ν

D→ t2|B|−1.

This method allows us to estimate the variance of an estimator and to quantify the uncer-

tainty in the variance estimator. In the testing procedures described in this paper we make

frequent use of this method. We describe below how it is used in the spatial stationarity test.

To implement the test, we define a set S ∈ Zd that surrounds but does not include

zero (examples include S = (1, 0), (1, 1), (0, 1), (−1, 1)) and test for stationarity using the

coefficients, Aλ(g; r); r ∈ S. Of course, the variance ca,λ is unknown and needs to be esti-

mated from the data. To estimate the variance, we observe from (8) that (a) <Aλ(g; r)

and =Aλ(g; r) have the same variance and (b) for all r ‘close’ to zero the variance of

<Aλ(g; r),=Aλ(g; r) is approximately the same which allows us to use the orthogonal

sample method described in Remark 2.2 to estimate the variance. Therefore, we define a

set S ′ ∈ Zd which is relatively ‘close’ to S, but S ∩ S ′ = ∅. We then estimate ca,λ using

ca,λ(S ′) := σ2(λd/2Aλ(g; r); r ∈ S ′) (where σ2(·) is defined in (9)). Using ca,λ(S ′) and (8)

we have

λd/2<Aλ(g; r)√ca,λ(S ′)

D→ Z1,r√1

2|S′|−1χ2

2|S′|−1

∼ t2|S′|−1 and λd/2=Aλ(g; r)√ca,λ(S ′)

D→ Z2,r√1

2|S′|−1χ2

2|S′|−1

∼ t2|S′|−1,

(10)

for r ∈ S with λd/n → 0 as n → ∞ and λ → ∞ (so called mixed domain asymptotics),

where Z1,r, Z2,r; r ∈ S are iid standard normal random variables and χ22|S′|−1 is a chi-

squared distributed random variable (with 2|S ′| − 1 degrees of freedom) which is the same

for all r ∈ S, but independent of Z1,r, Z2,r; r ∈ S, and tq denotes the t-distribution with

q degrees of freedom. A test statistic can then be defined as maxr∈S [|Aλ(g; r)|2/ca,λ(S ′)],whose limiting distribution can easily be obtained from (10). Note that a test statistic

based on the sum of squares rather than the maximum is also possible, however in terms of

simulations the maximum statistic tends to have slightly better power.

Just as in the nonstationary time series case, in order to obtain the limit of Aλ(g; r)

in the nonstationary spatial case, we use rescaled asymptotics. We define a sequence of

nonstationary spatial processes Zλ(s) (we use the term ‘sequence’ loosely, since λ is defined

on R+ and not on Z+), where for each λ > 0 and s ∈ [−λ/2, λ/2]d the covariance of Zλ(s)is

cov[Zλ(s), Zλ(s + v)] = κ(v;

s

λ

),

10

where κ : Rd × [−1/2, 1/2]d → R (note that s ∈ [−λ/2, λ/2]d) is the location-dependent co-

variance function. The corresponding location-dependent spectral density function is defined

as

F(Ω;

s

λ

)=

∫Rdκ(v;

s

λ

)exp(−i2πv′Ω)dv.

Under this set-up we have Aλ(g; r)P→ A(g; r) as λ→∞ where

A(g; r) =1

(2π)d

∫Rd

∫[−1/2,1/2]d

F (Ω;u) exp (−i2πu′r) g(Ω)dudΩ.

We observe that if in the test we let g(Ω) = exp(iv′Ω) then A(eiv·; r) is the Fourier coefficient

of∫

[−1/2,1/2]dκ(v;u) exp(−ir′u)du. Hence the test is geared towards detecting changes in

the covariance at lag v.

2.3 The test for temporal stationarity using orthogonal samples

Let us return to the temporal stationarity test discussed in Section 2.1. Suppose ` is fixed,

then by (3) and (4), we test for stationarity with CT (r, `); r ∈ T where T = 1, . . . ,mand use the L2-type statistic defined in (5).

In this section we apply the ideas in Section 2.2 to propose a different method for es-

timating v`(ωr). We recall that <CT (r, `) and =CT (r, `) have ‘variance’ v`(ωr) and that

v`(ωr) = v`(0) +O(|r|/T ). Define the set T ′ = m+B1, . . . ,m+B2 where B1 < B2. If B2

is not ‘too large’ then for r ∈ T ′ the ‘variance’ of CT (r, `) is approximately v`(0). Therefore

we use

v` := σ2(T 1/2CT (r, `); r ∈ T ′),

as an estimator of v` := v`(0), where σ2(·) is defined in (9). Based on this estimator we use

the test statistic

Tm = Tm∑r=1

|CT (r, `)|2

v`.

It can be shown that v`P→ v` as |T ′| → ∞, therefore if m is kept fixed we have that

TmD→ χ2

2m with (|T ′| + B1)/T → 0 as T ′ → ∞ and T → ∞. However, for finite B2,

i.e., |T ′| < ∞, a better finite sample approximation of the distribution of the test statistic

under the null uses (4) and that v` is asymptotically independent of∑m

r=1 |CT (r, `)|2 to give

11

T∑m

r=1 |CT (r, `)|2/v`D→∑2m

j=1 Z2j and v`/v`

D→ 12|T ′|−1

χ22|T ′|−1 such that we obtain

TmD→

∑2mj=1 Z

2j

12|T ′|−1

χ22|T ′|−1

,

where Zj2mj=1 are iid standard normal random variables which are independent of the χ2

random variable.

We note that to maximize power we require that |B1| → ∞ in such a way that |B1|/T → 0

as T →∞. To understand why, we observe that under the alternative CT (r, `) are estimating

Fourier coefficients (see (6)). If B1 is close to the origin then CT (r, `); r ∈ T ′ will be

estimating ‘large’ values, thus the sample variance of CT (r, `); r ∈ T ′ is likely to be large

thus reducing Tm and consequently the power of the test.

3 Properties of spatio-temporal Fourier transforms

We now use some of the ideas discussed in the previous section to test for stationarity of

a spatio-temporal process. Let us suppose that Zt(s); s ∈ Rd, t ∈ Z is a spatio-temporal

process which is observed at time t = 1, . . . , T and at locations sjnj=1 on the region

[−λ/2, λ/2]d. At any given time point, t, we may not observe all sjnj=1 locations, but

only a subset st,jntj=1, i.e., the data set we observe is Zt(st,j); j = 1, . . . , nt, t = 1, . . . , T.Throughout this paper we will use the following set of assumptions.

Assumption 3.1

(i) sj are iid uniformly distributed random variables on the region [−λ/2, λ/2]d.

(ii) The number of locations that are observed at each time point is nt, where for some

0 < c1 ≤ c2 <∞ (this does not change with n) we have c1n ≤ nt ≤ c2n.

(iii) The asymptotics are mixed, that is as λ→∞ (spatial domain grows), we have n→∞(number of locations grows) such that λd/n→ 0. We also assume that T →∞.

In much of the discussion below we restrict ourselves to the case r2 ∈ 0, 1, . . . , T/2 − 1,but allow r1 ∈ Zd.

Throughout the following, let Ωk = 2π(k1/λ, . . . , kd/λ), where k = (k1, . . . , kd) ∈ Zd

denote spatial frequencies and ωk = 2πk/T denote temporal frequencies. Keeping time or

location fixed, respectively, we define the Fourier transform over space at time t as

Jt(Ωk) =λd/2

nt

nt∑j=1

Zt(st,j) exp(is′t,jΩk),

12

and the Fourier transform over time at location sj as

Jsj(ωk) =1√2πT

T∑t=1

n

ntδt,jZt(sj)e

itωk ,

where, δt,j = 0 if at time t the location sj is not observed, otherwise δt,j = 1. Observe that the

ratio n/nt gives a large weight to time points where there are only a few observed locations.

We then define the spatio-temporal Fourier transform, i.e., the Fourier transformation over

space and time as

J(Ωk1 , ωk2) =1√2πT

T∑t=1

Jt(Ωk1) exp(itωk2) =λd/2

n

n∑j=1

Jsj(ωk2) exp(is′jΩk1). (11)

Our objective is to test for second order stationarity of the spatio-temporal process, in the

sense that cov[Zt(s), Zt+h(s + v)] = κh(v). In the remainder of this section we evaluate the

covariance cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)] under all combinations of temporal and spatial

stationarity and nonstationarity, respectively. This will motivate the testing procedures

proposed in Section 4.

We first note there is a subtle but important difference between the spectral density over

time and space. Under second order stationarity in space and time of Zt(s) the spectral

density is

f(Ω, ω) =1

2π

∑h∈Z

e−ihω∫Rdκh(v) exp(−iv′Ω)dv,

where the equality above is due to κh(v) = κ−h(−v). We observe that f : Rd × [0, 2π]→ R,

that is, f(·, ω) is defined on Rd (this is because the spatial process is defined over Rd) whereas

f(Ω, ·) is a periodic function defined on the interval [0, 2π] (because the temporal process is

over discrete time Z). The space-time spectral density of the type defined above is studied

in detail in Subba Rao and Terdik [2016].

To understand how the correlations between the Fourier transforms behave in the case

that the spatio-temporal process is nonstationary we will use the rescaling devise discussed

in Sections 2.1 and 2.2. To be able to apply the rescaling devise in space and time, we

assume that the ‘observed’ process Zt(s) is an element of a sequence (indexed over λ and T )

of nonstationary spatio-temporal processes Zt,λ,T (s); t ∈ Z, s ∈ Rd, i.e., Zt(s) = Zt,λ,T (s).

Using this formulation we can then place certain regularity conditions on the covariance. To

do so, we define the sequence ρh, such that∑

h |hρh| < ∞ and function βη(v), such that

13

for some η > 0, βη(v) =∏d

j=1 βη(vj) with

βη(vj) =

C |vj| ≤ 1

C|vj|−η |vj| > 1(12)

for some finite constant C. We assume there exists a time and location dependent spatio-

temporal covariance, κh;u : Rd × [−1/2, 1/2]d → R, such that for all T ∈ Z+, λ > 0, h ∈ Zand u ∈ [0, 1], we have

cov[Zt,λ,T (s), Zt+h,λ,T (s + v)] = κh; tT

(v;

s

λ

)+O

(ρhβ2+δ(v)

T

). (13)

The function κ·(·) satisfies the Lipschitz conditions: (i) supu,u |κh;u(v;u)| ≤ ρhβ2+δ(v), (ii)

supu |κh;u1 (v;u)−κh;u2 (v;u) | ≤ |u1−u2|ρhβ2+δ(v) and (iii) supu |κh;u (v;u1)−κh;u (v;u2) | ≤||u1 − u2||2ρhβ2+δ(v). Note that the index h; t/T refers to covariance at time lag h and

rescaled time t/T whereas (v; s/λ) refers to spatial covariance “lag” v and rescaled location

s/λ. Using the above definitions we define the time and location dependent spectral density

as

Fu(Ω, ω;u) =1

2π

∑h∈Z

e−ihω∫Rdκh;u(v;u)e−iv

′Ωdv. (14)

In the proposed testing procedure we also consider one-way stationarity tests, where we test

for stationarity over one domain without assuming stationarity over the other domain. To

understand how these tests behave, we use the following rescaling devises:

• Spatial stationarity and temporal nonstationarity

In this case, we assume that Zt(s) = Zt,T (s) and there exists a κ such that

cov[Zt(s), Zt+h(s + v)] = κh; tT

(v) +O(ρhβ2+δ(v)T−1).

The corresponding time-dependent spectral density is F tT

(Ω, ω) (defined analogously

to (14)).

• Temporal stationarity and spatial nonstationarity

In this case we assume Zt(s) = Zt,λ(s) and there exists a κ such that

cov[Zt(s), Zt+h(s + v)] = κh

(v;

s

λ

)with corresponding location dependent spectral density F (Ω, ω; s

λ).

14

In the following lemma we derive the properties of the DFT for the four different combi-

nations of temporal and spatial stationarity and nonstationarity, respectively.

Lemma 3.1 Suppose Assumption 3.1 is satisfied. We further assume that under spatial and

temporal stationarity |κh(v)| ≤ ρhβ2+δ(v), temporal stationarity supu |κh(v;u)| ≤ ρhβ2+δ(v),

spatial stationarity supu |κh;u(v)| ≤ ρhβ2+δ(v) and temporal and spatial nonstationarity

supu,u |κh;u(v;u)| ≤ ρhβ2+δ(v) with β2+δ(v) and ρh as defined in (12). Let b = b(r)

denote the number of zero entries in the vector r.

(i) If the process Zt(s); s ∈ Rd, t ∈ Z is spatially and temporally stationary, we have

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

=

f(Ωk1 , ωk2) +O

(1T

+ 1λ

+ λd

n

)r1 = 0 and r2 = 0

O(

1T

+ λd

n

)r1 = 0 and r2 6= 0

O(

1λd−b

)r1 6= 0 and r2 = 0

O(

1Tλd−b

)r1 6= 0 and r2 6= 0.

(ii) If the process is spatially stationary and temporally nonstationary, we have

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

=

∫ 1

0Fu(Ωk1 , ωk2) exp(−i2πr2u)du+O

(1T

+ λd

n

)r1 = 0 and r2 ∈ Z

O(

1λd−b

+ 1T

)r1 6= 0 and r2 ∈ Z

(iii) If the process is spatially nonstationary and temporally stationary, we have

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

=

∫[−1/2,1/2]d

exp(−i2πr′1u)F (Ωk1 , ωk2 ;u)du +O(

1T

+ 1λ

)r2 = 0 and r1 ∈ Zd

O(

1T

)r2 6= 0 and r1 ∈ Zd

(iv) If the process is spatially and temporally nonstationary, we have

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

=

∫ 1

0

exp(−i2πr2u)

∫[−1/2,1/2]d

exp(−i2πr′1u)Fu(Ωk1 , ωk2 ;u)dudu+O

(1

λ+

1

T+λd

n

).

In the above lemma we see that if the process is stationary then for non-zero values of r1 or

r2 the covariance between the DFTs is close to zero. On the other hand, when the process is

nonstationary the correlation is non-zero. In particular, cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

15

is approximately equal to the Fourier coefficient br2(Ωk1 , ωk2 ; r1), where

br2(Ω, ω; r1) =

∫ 1

0

exp(−i2πr2u)

∫[−1/2,1/2]d

exp(−i2πr′1u)Fu(Ω, ω;u)dudu. (15)

We note that Fu(Ω, ω;u) =∑

r1∈Zd∑

r2∈Z br2(Ω, ω; r1)e2πi(r′1u+r2u). Therefore, in the case

the spatio-temporal process is stationary, for all r1 6= 0 or r2 6= 0 we have br2(Ω, ω; r1) = 0

and for all u and u, Fu(Ω, ω;u) = b0(Ω, ω; 0) = f(Ω, ω) holds.

However, in the nonstationary case we have:

• Spatial stationarity, but temporal nonstationarity

For all r1 6= 0, br2(Ω, ω; r2) = 0. But for at least some r2 6= 0 and [Ω, ω] ∈ Rd × [0, 2π]

(measure non-zero), br2(Ω, ω; 0) 6= 0. In other words, the temporal nonstationarity is

‘seen’ on the r2-axis.

• Temporal stationarity, but spatial nonstationarity

For all r2 6= 0, br2(Ω, ω; r2) = 0. But for at least some r1 6= 0 and [Ω, ω] ∈ Rd × [0, 2π]

(measure non-zero), b0(Ω, ω; r1) 6= 0. In other words, the spatial nonstationarity is

‘seen’ on the r1-axis.

• Temporal and spatial nonstationarity

For at least some r1 6= 0 and r2 6= 0 and [Ω, ω] ∈ Rd × [0, 2π] (measure non-zero), we

have br2(Ω, ω; r1) 6= 0.

Using this dichotomy between stationary and nonstationary processes, our proposed test for

stationarity is based on estimates of br2(Ω, ω; r1). However, it is not feasible to test over

all (r1, r2) ∈ Zd+1. Instead, we note that since∫ 1

0

∫[−1/2,1/2]d

|Fu(Ω, ω;u)|2dudu < ∞, we

have∑

r1,r2|br2(Ω, ω; r1)|2 < ∞. Therefore |br2(Ω, ω; r1)| → 0 as ‖r1‖ → ∞ or |r2| → ∞.

Thus a test based on br2(Ω, ω; r1) should use (r1, r2) which are close to the origin (where

the deviations from zero are likely to be largest, thus leading to maximum power), from now

onwards we denote this test set as P = S × T .

For a given (r1, r2), one possibility is to simply estimate br2(Ω, ω; r1) for all Ω and ω.

Therefore, if br2(Ω, ω; r1) is non-zero for some values Ω, ω of non-zero measure, the test will

(asymptotically) have power. However, the drawback of a such an omnipresent test is that it

has very little power for small deviations from stationarity (i.e., when br2(Ω, ω; r1) is small).

Therefore in the following section we propose two different testing approaches. The first

estimates a weighted integral of br2(Ω, ω; r1), that is

Ag,h(r1, r2) = 〈br2(·, ·; r1), g(·)h(·)〉 =1

(2π)dπ

∫ π

0

∫Rdg(Ω)h(ω)br2(Ω, ω; r1)dΩdω,

16

for a given set of (weight) functions g : Rd → R and h : [0, π] → R. This test has the most

power for small deviations from stationarity - but they have to be in a direction that Ag,h

is non-zero. The second testing method is a compromise, between the omnipresent test and

the above test. In this test we estimate

Dg,h,v(r1, r2) =1

π

∫ π

0

v(ω)

[h(ω)

(2π)d

∫Rdg(Ω)br2(Ω, ω; r1)dΩ

]2

dω (16)

for a given set of functions g, h and v. This test uses g(Ω) to set the spatial features it

wants to detect, but the sum of squares over all frequencies ω means that it can detect for

deviations from temporal stationarity at all frequencies ω.

4 The spatio-temporal test for stationarity

In this section we focus on testing for stationarity of a spatio-temporal process. In Section

5 we adapt these methods to testing for one-way stationarity of a spatio-temporal process.

4.1 Measures of correlation in the Fourier transforms

Our aim is to test for second order stationarity by measuring the linear dependence between

the Fourier transforms. To do this, we recall that the test for spatial stationarity is a sum

of (weighted) sample autocovariances of J(Ωk) (see (7)). We now define an analogous

quantity to test for spatio-temporal stationarity. We start by defining the weighted sample

cross-covariance between J(Ωk1 , ωk2) and J(Ωk1+r1 , ωk2+r2) over k1 (but with k2 kept

fixed)

ag(ωk2 ; r1, r2) =1

λd

a∑k1=−a

g(Ωk1)J(Ωk1 , ωk2)J(Ωk1+r1 , ωk2+r2)−NT

=1

n2

a∑k1=−a

g(Ωk1)n∑

j1,j2=1

j1 6=j2

Jsj1 (ωk2)Jsj2 (ωk2+r2) exp(isj1Ωk1 − isj2Ωk1+r1), (17)

and g : Rd → R is a user chosen bounded Lipschitz continuous function (see Remark 2.1), a

is such that (aλ)d/n2 → 0, where the last line follows from (11) and

NT =1

n2

a∑k1=−a

g(Ωk1)n∑j=1

Jsj(ωk2)Jsj(ωk2+r2) exp(−isjΩr1) (18)

=1

2πT

a∑k1=−a

g(Ωk1)T∑

t,τ=1

eitωk2−iτωk2+r11

ntnτ

nt,τ∑j=1

Zt(st,τ,j)Zτ (st,τ,j) exp(−isjΩr1),

17

where nt,τ denotes the number of common locations at time t and τ . Our reason for removing

the term NT are two fold; the first is to remove the so called nugget effect which arises due

to measurement error in the spatial observations, the second reason is that NT tends to

inflate the variance of ag(·) (removing such a term is quite common in spatial statistics, see

Matsuda and Yajima [2009], Subba Rao [2015a] and Bandyopadhyay et al. [2015]).

Remark 4.1 An alternative choice of NT is

NT =1

2πT

a∑k1=−a

g(Ωk1)T∑t=1

exp(−itωr2)1

n2t

nt∑j=1

Z2t (st,j) exp(−isjΩr1).

Examples of weight functions g(·) are given in Remark 2.1. We will show in Lemma 4.1

that in many ways the sampling properties of λd/2ag(ωk2 ; r1, r2) resemble the temporal DFT

covariance JT (ωk)JT (ωk+r); compare Section 2.1. To prove this result we require the following

assumptions.

Assumption 4.1 Suppose Zt(u);u ∈ Rd, t ∈ Z is a fourth order stationary spatio-

temporal process. Let κh1,h2,h3(v1,v2,v3) = cum[Zt(s), Zt+h1(s+v1), Zt+h2(s+v2), Zt+h3(s+

v3)] and define the functions

fh(Ω) =

∫Rdκh(v) exp(−iv′Ω)du, and

fh1,h2,h3(Ω1,Ω2,Ω3) =

∫Rdκh1,h2,h3(v1,v2,v3) exp(−iv′1Ω1 − iv′2Ω2 − iv′3Ω3)dv1dv2dv3.

(i) fh(·) satisfies∫Rd |fh(Ω)|dΩ ≤ ρh,

∫Rd |fh(Ω)|2dΩ ≤ ρh and fh(Ω) ≤ ρhβ1+δ(Ω).

(ii) For all 1 ≤ j ≤ d, the partial derivatives satisfy |∂fh(Ω)∂Ωj| ≤ ρhβ1+δ(Ω), where Ω =

(Ω1, . . . ,Ωd).

(iii) |fh1,h2,h3(Ω1,Ω2,Ω3)| < ρh1ρh2ρh3∏d

j=1 β1+δ(Ω1,j)∏d

j=1 β1+δ(Ω2,j)∏d

j=1 β1+δ(Ω3,j) and

for 1 ≤ i ≤ 3 and 1 ≤ j ≤ d,∣∣∣∣∂fh1,h2,h3(Ω1,Ω2,Ω3)

∂Ωi,j

∣∣∣∣ ≤ ρh1ρh2ρh3

d∏j=1

β1+δ(Ω1,j)d∏j=1

β1+δ(Ω2,j)d∏j=1

β1+δ(Ω3,j).

In the results below we also require the fourth order spectral density

f4(Ω1, ω1,Ω2, ω2,Ω3, ω3) =1

(2π)3

∑h1,h2,h3∈Z

fh1,h2,h3(Ω1,Ω2,Ω3)e−ih1ω1−ih2ω3−ih3ω3 .

18

4.1.1 Sampling properties of ag(·) under stationarity

Below we derive the mean, variance and asymptotic normality of ag(·) from (17) under the

assumption that the spatio-temporal process is fourth order stationary.

Lemma 4.1 Suppose Assumptions 3.1 and 4.1 hold. In addition, | ∂dfh(Ω)

∂Ω1...∂Ωd| ≤ ρhβ1+δ(Ω)

(see the proof of Theorem 3.1, Subba Rao [2015a]). Then

E [ag(ωk; r1, r2)]

=

O(

1Tλd−b

∏d−bj=1 (log λ+ log |mj|)

)r1 ∈ Zd/0 and r2 6= 0

O(

1λd−b

∏d−bj=1 (log λ+ log |mj|)

)r1 ∈ Zd/0 and r2 = 0

O(

1T

)r1 = 0 and r2 6= 0

1(2π)d

∫Ω∈Rd g(Ω)f(Ω, ωk)dΩ +O

(log λλ

+ 1n

)r1 = 0 and r2 = 0

,

where b = b(r1) denotes the number of zeros in the vector r1 and mjd−bj=1 are the non-zero

values in r1.

Lemma 4.2 Suppose Assumptions 3.1 and 4.1 hold and r2, r4 are such that 0 ≤ r2, r4 ≤ T/2.

Then we have,

λdcov [<ag(ωk2 , r1, r2),<ag(ωk4 , r3, r4)]

= Ir1=r3Ir2=r4

[Ik2=k4Vg(ωk2 ; Ωr1 , ωr2) + Ik4=T−k2−r2Vg,2(ωk2 ; Ωr1 , ωr2) +O

(1

T

)]+O

(`λ,a,n

), (19)

where

Vg(ω; Ωr1 , ωr2) =1

2(2π)d

∫Dg(Ω)g(Ω)f(Ω, ω)f(Ω + Ωr1 , ω + ωr2)dΩ,

Vg,2(ω; Ωr1 , ωr2) =1

2(2π)d

∫Dr1

g(Ω)g(−Ω−Ωr1)f(Ω,−ω)f(−Ω−Ωr1 , ω + ωr2)dΩ,

and∫Dr1

=∫ 2πmin(a,a−r1,1)/λ

2πmax(−a,−a−r1,1)/λ. . .∫ 2πmin(a,a−r1,d)/λ

2πmax(−a,−a−r1,d)/λ. Note that `λ,a,n and

∫D are defined in

Section 2.2. Exactly the same result as in (19) holds for λdcov [=ag(ωk2 , r1, r2),=ag(ωk4 , r3, r4)],

whereas λdcov [<ag(ωk2 , r1, r2),=ag(ωk4 , r3, r4)] = O(`λ,a,n +Ir1=r3Ir2=r4

T).

Let (kj, r1, r2); 1 ≤ j ≤ m, (r1, r2) ∈ P and kj1 6= T − kj2 − r2 be a collection of integer

19

vectors. Then under under sufficient mixing conditions of Zt(s) we have

λd/2

[<ag(ωkj ; r1, r2)√Vg(ωkj ; Ωr1 , ωr2)

,=ag(ωkj ; r1, r2)√Vg(ωkj ; Ωr1 , ωr2)

; 1 ≤ j ≤ m, (r1, r2) ∈ P

]D→ N

(0, I2m|P|

),

as λd/n→ 0, n→∞, λ→∞ and T →∞.

We observe that for ‖r1‖2 << λ and |r2| << T and by the smoothness of the spectral density

f and tri-spectral density f4, we have

Vg(x1; Ωr1 , ωr2) = Vg(x2) +O (|x1 − x2|+ |Ωr1|+ |ωr2|) , (20)

where Vg(x) = Vg(x; 0, 0). We use these approximations in Sections 4.2.1 and 4.3.1.

The lemmas above show that ag(ω; r1, r2) is estimating zero in the case that the spatio-

temporal process is fourth order stationary. We observe that the variance of ag(ωk; r1, r2)

does not involve Vg,2(·). Therefore, in the definition of the test statistic, in Section 4.2 we

average ag(ωk; r1, r2) over the frequencies ωkT/2k=1. This is to avoid correlations between

ag(ωk; r1, r2) and ag(ωT−k−r2 ; r1, r2) and thus the need to estimate Vg,2.

In the section below we show that ag(ω; r1, r2) behaves differently in the case that the

spatio-temporal process is nonstationary.

4.1.2 Sampling properties of ag(·) under nonstationarity

Using the rescaled asymptotic set-up described in Section 3 and the assumptions in Lemma

3.1 we can show that under the alternative of nonstationarity

E [ag(ωk; r1, r2)] = bg,r2(ω; r1) +O

(λd

n+

1

λ+

1

T

),

where

bg,r2(ω; r1) = 〈g, br2(·, ω; r1)〉 =1

(2π)d

∫Rdg(Ω)br2(Ω, ω; r1)dΩ (21)

and br2(·, ω; r1) is defined in (15). Therefore, we see that if the process is nonstationary,

ag(ωk; r1, r2) is, in some sense, measuring the nonstationarity at frequency (r1, r2) in the

spectrum.

4.2 Test statistic 1: The average covariance

Motivated by the results above we define the average covariance. To do so, we first note

that Lemma 4.2 above shows that there is a ‘significant’ correlation between <ag(ωk; r1, r2)

and <ag(ωT−k−r2 ; r1, r2) (and likewise for the imaginary parts). Therefore we restrict the

20

summands below to the frequencies 1, . . . , T/2 to ensure the elements of the sum are mostly

near uncorrelated. We define the weighted sum as

Ag,h(r1, r2) =2

T

T/2∑k=1

h(ωk)ag(ωk; r1, r2), (22)

for given (user-chosen) weight functions g and h. The sampling properties of Ag,h(r1, r2) are

given in Lemma 6.1. Summarizing Lemma 6.1, the variances are

var

(√λdT

2<Ag,h(r1, r2)

)= Vg,h + o(1) and var

(√λdT

2=Ag,h(r1, r2)

)= Vg,h + o(1) (23)

and √λdT

2Vg,h

(<Ag,h(r1, r2),=Ag,h(r1, r2); (r1, r2) ∈ P

)D→ N (0, I2|P|) (24)

as `λ,a,n → 0 with λ→∞ and T →∞, where

Vg,h =1

2π

∫ π

0

|h(ω)|2Vg(ω)dω

+2

(2π)d+2

∫ π

0

∫ π

0

∫D2

g(Ω1)g(Ω2)h(ω1)h(ω2)f4(Ω1, ω1,Ω2, ω2,−Ω2,−ω2)

dΩ1dΩ2dω1dω2. (25)

Therefore, based on the above we can use an L2 or max norm as the test statistic, i.e.

λdT

2Vg,h

∑(r1,r2)∈P

|Ag,h(r1, r2)|2 orλdT

2Vg,hmax

(r1,r2)∈P|Ag,h(r1, r2)|2,

which is asymptotically either a chi-square statistic or the maximum of chi-squares with

2 degrees of freedom. However, we stumble across a problem in that the variance Vg,h is

generally unknown. One solution to this is to use the method of orthogonal samples described

in Section 2. We observe from (23) and (24) that if (r1, r2) is not too far from the origin,

then Ag,h(r1, r2); (r1, r2)’s asymptotically have the same variance and are uncorrelated.

Therefore, we can estimate the variance using the elements in the set P ′. We describe how

to construct the sets P and P ′ below.

Definition 4.1 (The sets P and P ′) We define the sets P = S × T and P ′ = S ′ × T ′,where S, S ′, T and T ′ are similar to the sets defined in Section 2.2 and Section 2.3 (S and

S ′ contain vectors in Zd whereas T and T ′ contain vectors in Z). We place the following

21

constraints on the sets; 0 /∈ P ,P ′, P ∩ P ′ = ∅. Furthermore if (r1, r2), (r3, r4) ∈ P or P ′,then (r1, r2) 6= −(r3, r4). P and P ′ are such that for (r1, r2) ∈ P ′, ‖r1‖2 << λ, |r2| << T .

P will be the set where we check for zero correlation and conduct the test and P ′ will be

the set which we use to estimate nuisance parameters. In order for the test statistics defined

below to be close to the nominal level, under the null of stationarity, the elements of P and

P ′ should be ‘close’ (in the sense of some distance measure). However, in order for the test

to have maximum power (i) the test set P should surround zero and (ii) if P ′ is too ‘close’

to P it can result in a loss of power. Further details can be found in Bandyopadhyay and

Subba Rao [2016].

Thus we estimate Vg,h with

Vg,h(P ′) = σ2

(√Tλd

2Ag,h(r1, r2); (r1, r2) ∈ P ′

). (26)

and use either the L2-statistic T1,g,h(P ,P ′) or the maximum statistic M1,g,h(P ,P ′) as the

test statistic, where

T1,g,h(P ,P ′) =λdT

2

∑(r1,r2)∈P

|Ag,h(r1, r2)|2

Vg,h(P ′)and M1,g,h(P ,P ′) =

λdT

2max

(r1,r2)∈P

|Ag,h(r1, r2)|2

Vg,h(P ′).

(27)

Asymptotically, under the null of stationarity we have

T1,g,h(P ,P ′)D→ χ2

2|P| and M1,g,h(P ,P ′)D→ F|P|,

as |P ′| → ∞, T →∞ and λ→∞, where F|P| is the distribution function of the maximum of

|P| i.i.d. exponentially distributed random variables with exponential parameter 1/2 (since

asymptotically under the null, (Tλd/2)|Ag,h(r1, r2)|2/Vg,h(P ′) limits to an exponential dis-

tribution) and is defined as F|P|(x) = |P|2

exp(−x/2)(1 − exp(−x/2))|P|−1. Using this result

we can test for stationarity at the α× 100%-level with α ∈ (0, 1).

Remark 4.2 (The test under nonstationarity) Suppose that Zt(s) = Zt,λ,T (s) is a non-

stationary spatio-temporal process. Then by using the rescaling devise defined in Section 3

we have Ag,h(r1, r2)P→ Ag,h(r1, r2) as T →∞, λd/n→ 0, λ→∞ and n→∞, where

Ag,h(r1, r2) =1

π(2π)d

∫ π

0

h(ω)

(∫Rdg(Ω)br2(Ω, ω; r1)dω

)dΩ,

and br2(Ω, ω; r1) is defined in (15).

22

We do not give the results of a formal local asymptotic analysis. However, suppose

µ(r1, r2) is a sequence where∑

r1,r2|µ(r1, r2)|2 <∞ and

Ag,h(r1, r2) =µ(r1, r2)

(Tλd)1/2.

If for some (r1, r2) ∈ P, µ(r1, r2) 6= 0, then the test will have power.

Of course in order to define the test statistic, we need to choose g and h. A reasonable

choice of g(·) is given in Remark 2.1. The choice of h is more complex and below we discuss

a choice of h that seems to give reasonable results in the simulations.

4.2.1 Choice of h

If we let h(ω) = exp(i`ω), then the test is designed to check for nonstationarity only in the

spatio-temporal covariance at temporal lag `, i.e., κ`;u(·; s). Instead, we use a weight function

similar to the temporal test described in Section 2.1, where we recall that in the construc-

tion of the temporal test statistic JT (ωk)JT (ωk+r)/

√fT (ωk)fT (ωk+r)’s are near uncorrelated

and CT (r, `) is pivotal in the case where the time series is stationary and Gaussian. Simi-

larly, in the construction of Ag,h(r1, r2) if we let h(ω) = Vg(ω)−1/2, where Vg(ω) is defined

in (20), we have <ag(ωk; r1, r2)/√Vg(ωk),=ag(ωk; r1, r2)/

√Vg(ωk are near uncorrelated,

asymptotically standard normal random variables. Thus we use h(ω) =√Vg(ω) to define

Ag,V −1/2(r1, r2) as

Ag,V −1/2(r1, r2) =2

T

T/2∑k=1

ag(ωk; r1, r2)√Vg(ωk)

,

which we see from (25) has variance

Vg,V −1/2 =1

2+

2

(2π)d+2

∫ π

0

∫ π

0

∫D2

g(Ω1)g(Ω2)√Vg(ω1)Vg(ω2)

f4(Ω1, ω1,Ω2, ω2,−Ω2,−ω2)

dΩ1dΩ2dω1dω2.

We observe from the above that in the case the spatio-temporal process is stationary and

Gaussian, Ag,V −1/2(r1, r2) is asymptotically pivotal; compare with temporal stationarity test

described in Section 2.2 where a similar result is true.

However, in general Vg(ω) is unknown and needs to be estimated. To estimate Vg(ωk)

we use the orthogonal sample method described in Remark 2.1 and the same set P ′ defined in

(4.1). Under these conditions we have that the real and imaginary parts of ag(ωk+i; r1, r2); (r1, r2) ∈P ′, |i| ≤ M for M << T , have almost the same variance and are near uncorrelated. Using

23

this we estimate Vg(ωk) with

Vg(ωk;P ′) = σ2(λd/2ag(ωk+i; r1, r2); (r1, r2) ∈ P ′, |i| ≤M

), (28)

where σ2(·) is defined in (9). We define the observed average covariance as

Ag,V −1/2(r1, r2) =2

T

T/2∑k=1

ag(ωk; r1, r2)√Vg(ωk;P ′)

.

By using the same methods described in Jentsch and Subba Rao [2015], Appendix A.2, we

can show that

λd/2T 1/2∣∣Ag,V −1/2(r1, r2)− Ag,V −1/2(r1, r2)

∣∣ P→ 0,

with |M |/T → 0 as M → ∞ and T → ∞. Hence Ag,V −1/2(r1, r2) and Ag,V −1/2(r1, r2) share

the same asymptotic sampling properties. Thus by using (24) we have√λdT

2Vg,V −1/2

(<Ag,V −1/2(r1, r2),=Ag,V −1/2(r1, r2); (r1, r2) ∈ P ∩ P ′

)D→ N (0, I2|P|). (29)

Since for a given data set, we cannot be sure if the underlying process is Gaussian, we estimate

the variance of Vg,V −1/2 using the method in (33) and use the test statistics T1,g,V −1/2(P ,P ′)and M1,g,V −1/2(P ,P ′) as defined in (27).

4.3 Test statistic 2: The average squared covariance

In the previous section we considered the average covariance for estimating the linear depen-

dence between the DFTs. As we can see from Remark 4.2 the average covariance is designed

to detect the frequency average deviation from stationarity. Of course by considering the

frequency average deviation, positive and negative frequency deviations can cancel leading

to an average deviation of zero, which would give the misleading impression of stationarity.

To address this issue we define a test statistic which estimates the average squared deviation

over all frequencies (and thus is designed to detect a wider range of alternatives). More

precisely, we group ag(ωk; r1, r2)T/2k=1 into blocks of length H and evaluate the local average

over each block

Bg,h;H(ωjH ; r1, r2) =1

H

H∑k=1

h(ωjH+k)ag(ωjH+k; r1, r2), for 0 ≤ j < T/(2H), (30)

24

where the length of block H is such that H/T as H →∞ and T →∞. For ease of notation

we assume that H is a multiple of T . This can be considered as a frequency localized version

of Ag,h(r1, r2) defined in the previous section. In Lemma 6.2 we show that√λdH

Wg,h(ωjH)

(<Bg,h;H(ωjH ; r1, r2),=Bg,h;H(ωjH ; r1, r2)

)D→ N (0, I2),

where Wg,h(ωjH) = Wg,h(ωjH ; 0, 0) with

Wg,h(ωjH)

=T

2Hπ

∫ ω(j+1)H

ωjH

|h(ω)|2Vg(ω)dω +T

2H(2π)2d+2

∫[ωjH ,ω(j+1)H ]2

∫D2

g(Ω1)g(Ω2)h(ω1)h(ω2)

×f4(Ω1, ω1,Ω2, ω2,−Ω2,−ω2)dΩ1dΩ2dω1dω2.

A careful examination of the expression above shows that the term involving the fourth order

cumulant is of lower order since it involves a double integral∫

[ωjH ,ω(j+1)H ]2= O((H/T )2). Thus

Wg,h(ωjH) =T

2Hπ

∫ ω(j+1)H

ωjH

|h(ω)|2Vg(ω; 0, 0)dω +O

(H

T

). (31)

Furthermore, the correlation between each of the blocks Wg,h(ωj1H) and Wg,h(ωj2H) is asymp-

totically negligible. Therefore, heuristically, we can treat the real and imaginary parts of

√

λdHWg,h(ωjH)

Bg,h;H(ωjH ; r1, r2); j = 0, . . . , T2H− 1 as ‘independent standard normal random

variables’ and define its mean squared average

Dg,h,W ;H(r1, r2) =2H

T

T/2H−1∑j=0

∣∣∣Bg,h;H(ωjH ; r1, r2)∣∣∣2

2Wg,h(ωjH).

Thus, E[Dg,h,W ;H(r1, r2)] = 1Hλd

and analogous to (24) we have√T

2H

[HλdDg,h,W ;H(r1, r2)− 1

]; (r1, r2) ∈ P

D→ N (0, I|P |), (32)

with H/T → 0 as H,T, λ → ∞. We define an L2 or maximum statistic based on the

above. However, in practice the variance Wg,h(ωjH) is unknown and once again we invoke

the method of orthogonal statistics to estimate it. We estimate Wg,h(ωj,H) with

Wg,h(ωjH ;P ′) = σ2(√

λdHBg,h(ωjH ; r1, r2); (r1, r2) ∈ P ′), (33)

25

and the observed Dg,h,W ;H(r1, r2) is defined with W in Dg,h,W ;H(r1, r2) replaced by W , that

is,

Dg,h,W ;H(r1, r2) =2H

T

T/2H−1∑j=0

∣∣∣Bg,h;H(ωjH ; r1, r2)∣∣∣2

2Wg,h(ωjH). (34)

The test statistic is constructed using the L2-sum

T2,g,h,W (P ,P ′) =

√T

2H

∑(r1,r2)∈P

HλdDg,h,W ;H(r1, r2),

and by using (32), under the of stationarity null, we have(T2,g,h,W (P ,P ′)−

√T

2H|P|)D→

N (0, |P|) with H/T → 0 and |P ′| → ∞ as T, λ,H → ∞. The maximum statistic M2,g,h,W

is defined analogously.

Remark 4.3 (The test under nonstationarity) Suppose that Zt(s) is a nonstation-

ary spatio-temporal process. Then by using the rescaling devise described in Section 3 we can

show that Dg,h,W ;H(r1, r2)P→ Dg,h,W ;H(r1, r2) as T → ∞, λd/n → 0, H/T → 0, H → ∞,

λ→∞ and n→∞, where Dg,h,W ;H(r1, r2) is defined in (16).

Again, without conducting a formal local asymptotic analysis, if

Dg,h,W ;H(r1, r2) =µ(r1, r2)

T 1/2H1/2λd

where∑

r1,r2|µ(r1, r2)|2 < ∞ and for some (r1, r2) ∈ P, µ(r1, r2) 6= 0, then the test will

have power.

4.3.1 Choice of h

Motivated by Section 4.2.1 we let h(ω) = Vg(ω;P ′)−1/2 and define the local average

Bg,V −1/2;H(ωjH ; r1, r2) =1

H

H∑k=1

ag(ωjH+k; r1, r2)√Vg(ωjH+k;P ′)

. (35)

By using (31) we see its real and imaginary parts have limiting variance

Wg,V −1/2(ωjH) = 1 +O

(H

T

). (36)

26

Therefore, we observe that by using h(ω) =√Vg(ω), Wg,V −1/2(ωjH) is asymptotically piv-

otal (even if the underlying spatio-temporal process is nonstationary). In other words, we

can treat the real and imaginary parts of √

2λdHBg,h;H(ωjH ; r1, r2); j = 0, . . . , T2H− 1 as

‘independent standard normal random variables’ and define its mean squared average

Dg,V −1/2,1;H(r1, r2) =2H

T

T/2H−1∑j=0

|Bg,V −1/2;H(ωjH ; r1, r2)|2

2. (37)

Studying D, we have avoided estimating the variance of Bg,V −1/2;H(ωjH ; r1, r2) by simply

replacing this variance by its limiting variance which is 1. In the simulation study in Section

7 we compare the effect this has on the finite sample properties of the test statistic. Using

(32) we have √T

2H

(HλdDg,V −1/2,1;H(r1, r2)− 1

)D→ N (0, 1) (38)

with H/T → 0 as T →∞, H →∞ and λ→∞. Therefore we define the test statistic

T2,g,V −1/2,1(P ,P ′) =

√T

2H

∑(r1,r2)∈P

HλdDg,V −1/2,1;H(r1, r2)

and that under the null of stationarity(T2,g,V −1/2,1(P ,P ′)− |P|

√T/2H

)D→ N (0, |P|).

Since approximately TλdDg,V −1/2,1;H(r1, r2) ∼ χ2T/H , chi-squared with T/H-degrees of free-

dom, a similar result can be derived for the analogous maximum statistic M2,g,V −1/2,1(P ,P ′)based on the maximum of chi-squares.

In Section 7 we compare T2,g,V −1/2,W (P ,P ′) and M2,g,V −1/2,W (P ,P ′) (when we standardize

with sample variance W ) with T2,g,V −1/2,1(P ,P ′) and M2,g,V −1/2,1(P ,P ′).

4.4 Asymptotic ‘finite sample’ approximations of the distribution

of the test statistics under the null

We recall that in Section 2.2 real and imaginary parts of the estimator Aλ(g;r)√ca,λ(S′)

converge

to a standard normal distribution under the null of stationarity as λ → ∞ and |S ′| → ∞.

However, in reality |S ′| is finite and not that large. Therefore, in (10) we estimate it with

the t-distribution, which can be considered as the ‘asymptotic finite sample distribution’ of

27

this ratio. In this section we use an analogous method to approximate the distributions of

|Ag,V −1/2(r1, r2)|2

Vg(P ′), Dg,V −1/2,W ;H(r1, r2) and Dg,V −1/2,1;H(r1, r2).

Let ZR,j(r1, r2), ZI,j(r1, r2); j = 1, . . . , T/2 and Zj,k; j = 1, . . . , 2|P ′|, k = 1, . . . , T/2denote iid standard Gaussian random variables (we use a double index because it simplifies

some of the notations later on). We recall from the definition of Vg(ωk;P ′) in (28) that it is

composed of (2M + 1)-local average of <ag(·) and =ag(·), each term being asymptotically

normal. Therefore we replace all the <ag(·) and =ag(·) in the definition of Vg(ωk;P ′) with

standard normal distributions to give

λd/2<ag(ωk; r1, r2)√

Vg(ωk;P ′)∼ tR,k(r1, r2) λd/2

=ag(ωk; r1, r2)√Vg(ωk;P ′)

∼ tI,k(r1, r2),

where

tR,k(r1, r2) = λd/2<ag(ωk; r1, r2)√

Vg(ωk;P ′)∼ ZR,k(r1, r2)√

12(2M+1)|P ′|−1

∑Mi=−M

∑2|P ′|j=1 (Zj,k+i − Zk)2

tI,k(r1, r2) = λd/2=ag(ωk; r1, r2)√

Vg(ωk;P ′)∼ ZI,k(r1, r2)√

12(2M+1)|P ′|−1

∑Mi=−M

∑2|P ′|j=1 (Zj,k+i − Zk)2

and Zk = 12(2M+1)|P ′|

∑Mi=−M

∑2|P ′|j=1 Zj,k+i is the local average.

Noting that the test statistic is in terms of λdT2|Ag,V −1/2(r1, r2)|2, we replace the real and

imaginary parts of ag(ωk; r1, r2)/

√Vg(ωk;P ′) with the above to give

Tλd

2

|Ag,V −1/2(r1, r2)|2

Vg,V −1/2

∼ Xr1,r2

where

Xr1,r2 =T

2

∣∣∣∣∣∣T/2∑k=1

tR,k(r1, r2)

∣∣∣∣∣∣2

+T

2

∣∣∣∣∣∣T/2∑k=1

tI,k(r1, r2)

∣∣∣∣∣∣2

,

for other (r1, r2), we use independent Zk,R(r1, r2), Zk,I(r1, r2) but the same Zj,k. We

also recall that we estimate the variance Vg,V −1/2 , therefore we approximate its distribution

with a weighted chi-squared with (2|P ′|−1) degrees of freedom. Since the orthogonal sample

28

which was used to estimate it contained 2|P ′| terms

Vg,V −1/2

Vg,V −1/2

∼ 1

2|P ′| − 1χ2

2|P ′|−1.

Therefore, based on the above, the following distribution is used

T1,g,V −1/2(P ,P ′) ∼1

12|P ′|−1

χ22|P ′|−1

∑(r1,r2)∈S1

Xr1,r2

to approximate the ‘asymptotic finite sample properties’ of T1,g,V −1/2(P ,P ′), under the

null. Using the same method we can obtain the ‘asymptotic finite sample properties’ for

M1,g,V −1/2(P ,P ′).Next we consider the local average DFTs, Bg,V −1/2(ωjH ; r1, r2), Dg,V −1/2,W ;H(r1, r2) and

Dg,V −1/2,1;H(r1, r2), which lead to the test statistics T2,g,V −1/2,W and T2,g,V −1/2,1. Using the

arguments given above we have,

Hλd∣∣∣Bg,V −1/2;H(ωjH ; r1, r2)

∣∣∣2 ∼ YjH(r1, r2),

where,

YjH(r1, r2) =

∣∣∣∣∣ 1√H

H∑k=1

tR,jH+k(r1, r2)

∣∣∣∣∣2

+

∣∣∣∣∣ 1√H

H∑k=1

tI,jH+k(r1, r2)

∣∣∣∣∣2

.

Using the above, we estimate the distribution of Dg,V −1/2,1;H(r1, r2) with

HλdDg,V −1/2,1;H(r1, r2) ∼ 2H

T

T/2H∑j=1

YjH(r1, r2).

This gives us the ‘asymptotic finite sample’ distributions of T2,g,V −1/2,1(P ,P ′) and M2,g,V −1/2,1(P ,P ′).In order to derive the sampling properties of T2,g,V −1/2,W (P ,P ′) and M2,g,V −1/2,W (P ,P ′),

we recall that Dg,V −1/2,W ;H(r1, r2) involves Wg,V −1/2(ωjk;P ′) and we approximate this distri-

bution by independent chi-squares 12|P ′|−1

χ22|P ′|−1,k

T/2Hk=1 . This gives

λdHDg,V −1/2,W ;H(r1, r2) ∼ 2H

T

T/2H∑j=1

YjH(r1, r2)1

2|P ′|−1χ2

2|P ′|−1,k

.

Using this we can obtain the distributions of T2,g,V −1/2,W (P ,P ′) and M2,g,V −1/2,W (P ,P ′).Note that the same 1

2|P ′|−1χ2

2|P ′|−1,kT/2Hk=1 is used for all Dg,V −1/2,W ;H(r1, r2); (r1, r2) ∈ P.

29

The ‘asymptotic finite sample’ distribution derived above are used in all the simulations

below.

5 Testing for one-way stationarity

In this section we gear the procedure to specifically test for stationarity over one domain,

without necessarily assuming stationarity over the other domain. For the one-way station-

arity test we use the same test statistics defined in Section 4, however we use observations

made in Section 3 in order to define the test set P over which the test statistic is defined.

For ag(·; r1, r2) as defined in (17) we observe:

• Spatial stationarity, but not necessarily temporal stationarity

If r1 6= 0 for all ω, we have ag(ω; r1, 0) = op(1). On the other hand, if the process is

spatially nonstationary then the latter is not necessarily true. Therefore the test set is

P = S × 0.

• Temporal stationarity, but not necessarily spatial stationarity

If r2 6= 0 for all ω we have ag(ω; 0, r2) = op(1). On the other hand, if the process is

temporally nonstationary then the latter is not necessarily true. Therefore the test set

is P = 0 × T .

We recall that in order to ensure the test statistics defined in Section 4 are asymptoti-

cally pivotal we used the method of orthogonal samples to estimate the variance for various

parts of the test statistic. Therefore, we need to ensure the set P ′ over which the orthogonal

sample is defined is such that it consistently estimates the variance. To do this we derive

expressions for the covariances of ag(ωk1 ; r1, r2) and Ag,h(r1, r2), respectively, under the gen-

eral nonstationary setting. In the following sections we consider the specific cases of spatial

or temporal stationarity.

By using (15) we can show that under temporal and spatial nonstationarity the covariance

of ag(ωk2 ; r1, r2) is

λdcov [<ag(ωk2 ; r1, r2),<ag(ωk4 ; r3, r4)]

=1

2<[b

(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3) + b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)]

+O

(1

T+

1

λ+ `λ,a,n

),

(39)

λdcov [=ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)]

=1

2<[b

(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)− b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)]

+O

(1

T+

1

λ+ `λ,a,n

),

30

and

λdcov [<ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)]

= −1

2=[b

(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)− b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)]

+O

(1

T+

1

λ+ `λ,a,n

),

where

b(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3) =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)×[bk4−k2(Ωk1 , ωk2 ;k3 − k1)bk4+r4−k2−r2(−Ωk1+r1 ,−ωk2+r2 ;k3 + r3 − k1 − r1)

+b−k4−k2−r4(Ωk1 , ωk2 ;−k3 − k1 − r3)bk4+k2+r2(−Ωk1+r1 ,−ωk2+r2 ;k1 + k3 + r1)

],

and

b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3) =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)[b−k4−k2(Ωk1 , ωk2 ;−k3 − k1)bk4+k2+r4+k2(−Ωk1+r1 ,−ωk2+r2 ;k3 + r3 + k1 + r1)

+bk4−k2+r4(Ωk1 , ωk2 ;k3 − k1 + r3)b−k4+k2+r2(−Ωk1+r1 ,−ωk2+r2 ;−k3 + k1 + r1)

].

Similar expressions for Ag,h(r1, r2) can be found in Section 6.2.

The above expressions are cumbersome, however, under one-way stationarity simplifica-

tions can be made. We recall from the definition in (15) that

br2(Ω, ω; r1) =

0 r1 6= 0 and spatial stationarity

0 r2 6= 0 and temporal stationarity(40)

We use these results to simplify the expressions for cov[ag(ωk2 ; r1, r2), ag(ωk4 ; r3, r4)] in the

case of one-way stationarity.

5.1 Testing for spatial stationarity

In this section we adapt the test to testing for spatial stationarity. By using (39) and

(40), under the null that Zt(s); t ∈ Z, s ∈ Rd is spatially stationary but not necessarily

31

temporally stationary we have

λdcov[<ag(ωk2 ; r1, r2),<ag(ωk4 ; r3, r4)] =

b

(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r1) +O( 1T

+ `λ,a,n) r1 = r3

b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1,−r1) +O( 1T

+ `λ,a,n) r1 = −r3

O( 1T

+ `λ,a,n) otherwise

where

b(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r1)

=1

2λd

a∑k1=−a

|g(Ωk1)|2bk4−k2 (Ωk1 , ωk2 ; 0) bk4+r4−k2−r2 (−Ωk1+r1 ,−ωk2+r2 ; 0)

+1

2λd

min(a,a+r1)∑k1=max(−a,−a−r1)

g(Ωk1)g(−Ωk1+r1)b−k2−k4−r4 (Ωk1 , ωk2 ; 0) bk2+k4+r4 (−Ωk1+r1 ,−ωk2+r2 ; 0) ,

and b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1,−r1) is defined similarly. The same result can be shown for

λdcov[<ag(ωk2 ; r1, r2)<ag(ωk4 ; r3, r4)]. Furthermore, λdcov[<ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)] =

o(1).

In order to test for spatial stationarity, without the temporal effect influencing the result,

we focus on r2 = 0. In this case, the above reduces to

λdcov[<ag(ωk2 ; r1, 0),<ag(ωk4 ; r3, 0)] =

b

(1)k2,k4

(ωk2 , ωk4 ; r1) +O(‖r1‖1λ

+ 1T

+ `λ,a,n) r1 = r3

b(2)k2,k4

(ωk2 , ωk4 ; r1) +O(‖r1‖1λ

+ 1T

+ `λ,a,n) r1 = −r3

O( 1T

+ `λ,a,n) otherwise

where

b(1)k2,k4

(ωk2 , ωk4 ; r1)

=1

2λd

a∑k1=−a

|g(Ωk1)|2bk4−k2 (Ωk1 , ωk2 ; 0) bk4+k2 (−Ωk1 ,−ωk2 ; 0)

+1

2λd

min(a,a+r1)∑k1=max(−a,−a−r1)

g(Ωk1)g(−Ωk1)b−k2−k4 (Ωk1 , ωk2 ; 0) bk2+k4 (−Ωk1 ,−ωk2 ; 0)

Furthermore, defining Ag,h(r1, 0) as in (22) we have

Tλd

2cov[<Ag,h(r1, 0),<Ag,h(r3, 0)]

≈

12b+O

(‖r1‖1λ

+ 1T

+ `λ,a,n

)r1 = r3

O(

1T

+ `λ,a,n)

otherwise, except when r1 = −r3.

32

where,

b =2

T

T/2∑k2,k4=1

h(ωk2)h(ωk4)b(1)k2,k4

(ωk2 , ωk4 ; r1) +

4

T 2λ2d

T/2∑k2,k4=1

a∑k1,k3=−a

h(ωk2)h(ωk4)g(Ωk1)g(Ωk3)b0,4(Ωk1 , ωk2 ,Ωk3 , ωk3 , ωk2 ,−Ωk3 ,−ωk3 ; 0)

and b0,4 is defined in Section 6.2.

Based on the above observations we define the test set P = S × 0 (where S surrounds

zero, but is such that if r1, r3 ∈ S then r1 6= −r3). The set over which the orthogonal

statistics are defined is P ′ = S ′ × 0, with S ∩ S ′ = ∅. The DFT covariance is defined as

Ag,V −1/2(r1, 0) =2

T

T/2∑k=1

ag(ωk; r1, 0)√Vg(ωk;P ′)

, where Vg(ωk;P ′) = σ2(λd/2ag(ωk; r1, 0); r1 ∈ S ′

).

We observe that unlike the spatio-temporal test described in Section 4, in the definition of

Vg(ωk;P ′) we only use frequency ωk (i.e., we should let M = 0).

We use Ag,V −1/2(r1, 0), defined above, to define the test statistics T1,g,h(P ,P ′) and

M1,g,h(P ,P ′) (see Section 4.2). Note that when testing for spatial stationarity, we have

to be careful about using the test statistics T2,g,V −1/2,W (P ,P ′) and T2,g,V −1/2,1(P ,P ′). This

is because when the process is temporally nonstationary the local average Bg,h;H(ωjH ; r1, r2)

is dependent over j.

5.2 Testing for temporal stationarity

Next we consider how to adapt the procedure to test for temporal stationarity. Under the

null that Zt(s) is temporally stationary but not necessarily spatially stationary and using

(39) and (40) we have,

λdcov[<ag(ωk2 ; r1, r2),<ag(ωk4 ; r3, r4)]

=

b

(1)r2,r2,k2,k2

(ωk2 , ωk2 ; r1, r3) +O(

1T

+ `λ,a,n)

r2 = r4, k2 = k4

O(

1T

+ `λ,a,n)

otherwise,

where, r2, r4, k2 and k4 is constrained such that 1 ≤ r2, r4, k2, k4 ≤ T/2 and

b(1)r2,r2,k2,k4

(ωk2 , ωk2 ; r1, r3)

=1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)b0(Ωk1 , ωk2 ;k3 − k1)b0(−Ωk1+r1 ,−ωk2+r2 ;k3 + r3 − k1 − r1)

33

A similar result holds for λdcov[=ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)] and cross-covariance term

λdcov[<ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)] is asymptotically zero.

In order to test for temporal stationarity and to avoid the influence of the spatial com-

ponent we focus on r1 = 0. In this case the above reduces to

λdcov[<ag(ωk2 ; 0, r2),<ag(ωk4 ; 0, r4)] =

12b(1)(ωk2) +O

(1+|r2|T

+ `λ,a,n

)r2 = r4, k2 = k4

O(

1T

+ `λ,a,n)

otherwise,

where,

b(1)(ωk2) =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)|b0(Ωk1 , ωk2 ;k3 − k1)|2.

And similarly for λdcov[=ag(ωk2 ; r1, r2),=ag(ωk4 ; r3, r4)]. Furthermore, defining Ag,h(0, r2)

as in (22) we have,

Tλd

2cov[<Ag,h(0, r2),<Ag,h(0, r4)] =

12c+O

(1+|r2|T

+ `λ,a,n

)r2 = r4

O(

1T

+ `λ,a,n)

otherwise,

where,

c =2

T

T/2∑k2=1

|h(ωk2)|2|b(1)(ωk2)|2 +4

T 2λ2d

T/2∑k2,k4=1

a∑k1,k3=−a

h(ωk2)h(ωk4)g(Ωk1)g(Ωk3)

×b0,4(Ωk1 , ωk2 ,Ωk3 , ωk3 , ωk2 ,−Ωk3 ,−ωk3 ; 0).

Using these observations we use the same test statistics as those described in Section 4.

The only differences are that we set r1 = 0 when we test for spatial stationarity and use

the set P = 0 × T (where T ⊂ Z+). We do the same in order to estimate the nuisance

parameters Vg(ω) and Vg,V −1/2 and Wg,V −1/2(ω) (where T ′ ⊂ Z+).

6 Auxiliary Results

6.1 Results in the case of stationarity

We first consider the sampling properties of Ag,h(r1, r2), which is used to define the test

statistics T1,g,V −1/2(P ,P ′) and M1,g,V −1/2(P ,P ′).

Lemma 6.1 Suppose Assumptions 3.1 and 4.1 hold, 0 ≤ r2, r4 ≤ T/2 − 1, r1 6= −r3 and

34

h : [0, π]→ R is a Lipschitz continuous function. Then

λdT

2cov

[<Ag,h(r1, r2),<Ag,h(r3, r4)

]= Ir1=r3Ir2=r4Vg,h(Ωr1 , ωr2) +O

(`λ,a,n + T−1

)(41)

λdT

2cov

[=Ag,h(r1, r2),=Ag,h(r3, r4)

]= Ir1=r3Ir2=r4Vg,h(Ωr1 , ωr2) +O

(`λ,a,n + T−1

)and λdT cov[<Ag,h(r1, r2),=Ag,h(r3, r4)] = O(`λ,a,n + T−1) where,

Vg,h(Ωr1 , ωr2) =1

π

∫ π

0

|h(ω)|2Vg(ω; Ωr1 , ωr2)dω +2

(2π)dπ2

∫ π

0

∫ π

0

∫D2

g(Ω1)g(Ω2)h(ω1)h(ω2)

×f4(Ω1 + Ωr1 , ω1 + ωr2 ,Ω2, ω2,−Ω2 −Ωr1 ,−ω2 − ωr2)dΩ1dΩ2dω1dω2.

Let (rj, rj); 1 ≤ j ≤ m be a collection of integer vectors constrained such that 0 ≤ rj ≤T/2−1 and rj1 6= −rj2. Then under stationarity of Zt(s) and sufficient mixing conditions

we have,√λdT

2

[<Ag,h(rj1 , rj1)

Vg,h(Ωrj1, ωrj1 )1/2

,=Ag,h(rj1 , rj1)

Vg,h(Ωrj1, ωrj1 )1/2

, . . . ,<Ag,h(rjm , rjm)

Vg,h(Ωrjm, ωrjm )1/2

,=Ag,h(rjm , rjm)

Vg,h(Ωrjm, ωrjm )1/2

]D→ N (0, I2m) .

We note that when ‖r1‖1 << λ and |r2| << T that the variances above approximate to

Vg,h(Ωr1 , ωr2) = Vg,h(0, 0) +O

(‖r1‖1

λ+|r2|T

).

We now consider the sampling properties of Bg,h;H(r1, r2) and Dg,h,v;H(r1, r2), which are

used to define the test statistics T2,g,V −1/2,W (P ,P ′) and M2,g,V −1/2,W (P ,P ′). We start by

studying Bg,h;H(r1, r2).

Lemma 6.2 Suppose Assumptions 3.1 and 4.1 hold, 0 ≤ r2, r4 ≤ T/2 − 1, r1 6= −r3 and

h : [0, π]→ R is a Lipschitz continuous function. Then

λdHcov[<Bg,h;H(ωjH ; r1, r2),<Bg,h;H(ωjH ; r3, r4)

]= Ir1=r3Ir2=r4Wg,h(ωjH ; Ωr1 , ωr2) +O

(`λ,a,n +H−1

). (42)

35

Exactly the same result holds for λdHcov[=Bg,h;H(ωjH ; r1, r2),=Bg,h;H(ωjH ; r3, r4)

], where

2Wg,h(ωjH ; Ωr1 , ωr2)

=T

Hπ

∫ ω(j+1)H

ωjH

|h(ω)|2Vg(ω; Ωr1 , ωr2)dω +T

H(2π)2d+2

∫ ω(j+1)H

ωjH

∫ ω(j+1)H

ωjH

∫D2

g(Ω1)g(Ω2)

×h(ω1)h(ω2)f4(Ω1 + Ωr1 , ω1 + ωr2 ,Ω2, ω2,−Ω2 −Ωr1 ,−ω2 − ωr2)dΩ1dΩ2dω1dω2,

noting that the first (covariance) term in Wg,h = O(1), whereas the second term of Wg,h which

is the fourth order cumulant term is of order O(H/T ) since the cumulant term involves a

double integral which is of order O((H/T )2). On the other hand, if (r1, r2), (r3, r4) 6= 0, then

(with 0 ≤ r2, r4 < T/2 and (r1, r2) 6= −(r3, r4)) we have λdHcov[<Bg,h;H(ωjH ; r1, r2),=Bg,h;H(ωjH ; r3, r4)

]=

O(`λ,a,n) and for j1 6= j2,

λdHcov[<Bg,h;H(ωj1H ; r1, r2),<Bg,h;H(ωj2H ; r3, r4)

]= O

(`λ,a,n +H−1 +

H

T

), (43)

where the same holds for λdHcov[=Bg,h;H(ωj1H ; r1, r2),=Bg,h;H(ωj2H ; r3, r4)

]and also for

λdHcov[<Bg,h;H(ωj1H ; r1, r2),=Bg,h;H(ωj2H ; r3, r4)

].

Let (kj, r1,i, r2,i); 1 ≤ j ≤ m, (r1, r2) ∈ P be a collection of integer vectors constrained

such that, 1 ≤ kj ≤ T/2, 1 ≤ rj ≤ T/2 and rj1 6= −rj2. Then under stationarity of Zt(s)and sufficient mixing conditions we have,

√λdH

[<Bg,h(ωkjH ; r1,i, r2,i)

Wg,h(ωkjH)1/2,=Bg,h(ωkjH ; r1,i, r2,i)

Wg,h(ωkjH)1/2, 1 ≤ j ≤ m, (r1,i, r2,i) ∈ P

]D→ N

(0, I2m|P|

),

where Wg,h(ω) = Wg,h(ω; 0, 0).

In the following lemma we consider the sampling properties of Dg,h,v;H(r1, r2). Note that

we consider general functions v, whereas in Section 4.3 we set v to be the variance of Wg,h(ω),

which means the mean of D is asymptotically pivotal.

Lemma 6.3 Suppose the assumptions in Lemma 6.2 hold and h : [0, π] → R is a Lipschitz

continuous function. Then we have

E[HλdDg,h,v;H(r1, r2)]

= Eg,h,v(Ωr1 , ωr2) +O

(`λ,a,n +

1

H+H

T+λdH[

∏d−bj=1 (log λ+ log |mj|)]2

(T Ir2−r4 6=0λd−b)2

)(44)

36

and

T

2Hcov

[λdHDg,h,v;H(r1, r2), λdHDg,h,v;H(r3, r4)

]=

Ug,h,v(Ωr1 , ωr2) +O

(HT

+ `λ,a,n)

r1 = r3 and r2 = r4

O(HT

+ `λ,a,n)

otherwise(45)

where,

Eg,h,v(Ωr1 , ωr2) =1

π

∫ π

0

Wg,h(ω; Ωr1 , ωr2)

v(ω)dω

and Ug,h,v(Ωr1 , ωr2) =1

π

∫ π

0

|Wg,h(ω; Ωr1 , ωr2)|2

|v(ω)|2dω.

Let (rj, rj); 1 ≤ j ≤ m be a collection of integer vectors constrained such that 1 ≤ kj ≤T/2, 1 ≤ rj ≤ T/2 and rj1 6= −rj2, then we have

√T

2HUg,h,v

λdHDg,h,v;H(rj1 , rj1)− Eg,h,v...

λdHDg,h,v;H(rjm , rjm)− Eg,h,v

D→ N (0, I2m)

where Ug,h,v = Ug,h,v(0, 0) and Eg,h,v = Eg,h,v(0, 0).

6.2 Results in the case of nonstationarity

We first generalize (13) from covariances to fourth order cumulants. We assume there exists

a function κ such that

cov[Zt,λ,T (s), Zt+h1,λ,T (s + v1), Zt+h2,λ,T (s + v2), Zt+h3,λ,T (s + v3)]

= κh1,h2,h3; tT

(v1,v2,v3;

s

λ

)+O

(∏3i=1 β2+δ(vi)ρhi

T

), (46)

where, supu,s |κh1,h2,h3;u(v1,v2,v3; s)| ≤∏3

i=1 ρhiβ2+δ(vi),

|κh1,h2,h3;u1 (v1,v2,v3; s)− κh1,h2,h3;u2 (v1,v2,v3; s) | ≤ |u1 − u2|3∏i=1

β2+δ(vi)ρhi ,

|κh1,h2,h3;u (v1,v2,v3; s1)− κh1,h2,h3;u (v1,v2,v3; s2) | ≤ |s1 − s2|13∏i=1

β2+δ(vi)ρhi .

37

Using the above, we define the location and time dependent fourth order spectral density as

Fu,4 (Ω1, ω1,Ω2, ω2,Ω3, ω3; s)

=1

(2π)3

∑h1,h2,h3∈Z

e−i(h1ω1+h2ω2+h3ω3)

∫R3d

κh1,h2,h3;u (v1,v2,v3; s)

×e−i(v′1Ω1+v′2Ω2+v′3Ω3)dv1dv2dv3. (47)

In order to obtain the expressions below we start by generalizing the covariance result in

(15) to fourth order cumulants. By using Lee and Subba Rao [2015] and similar methods to

those used in Bandyopadhyay and Subba Rao [2016] it can be shown that

cum[J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2), J(Ωk3 , ωk4), J(Ωk3+r3 , ωk4+r4)]

=1

Tλdbr2−r4,4(Ωk1+r1 , ωk2+r2 ,Ωk3 , ωk4 ,Ω−k3−r3 , ω−k4−r4 ; r1 − r3) +O

(1

T 2λd+

1

Tλd+1

)where,

br2,4(Ω1, ω1,Ω2, ω2,Ω3, ω3; r1)

=

∫ 1

0

∫[−1/2,1/2]d

Fu,4 (Ω1, ω1,Ω2, ω2,Ω3, ω3; s) e−2πir′1se−2πir2udsdu,

and Fu,4 is defined in (47).

Lemma 6.4 Suppose the assumptions in Assumptions 3.1 and 4.1 (generalized to the non-

stationary set-up) and (46) are satisfied. Then we have,

λdcov [ag(ωk2 ; r1, r2), ag(ωk4 ; r3, r4)] = b(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3) +O

(1

T+

1

λ

),

and

λdcov[ag(ωk2 ; r1, r2), ag(ωk4 ; r3, r4)

]= b

(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3) +O

(1

T+

1

λ+ `λ,a,n

).

38

We also have,

Tλd

2cov

[Ag,h(r1, r2), Ag,h(r3, r4)

]=

2

Tλd

T/2∑k2,k4=1

h(ωk2)h(ωk4)b(1)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)

+2

T 2λ2d

T/2∑k2,k4=1

a∑k1,k3=−a

h(ωk2)h(ωk4)g(Ωk1)g(Ωk3)

×br2−r4,4(Ωk1+r1 , ωk2+r2 ,Ωk3 , ωk4 ,Ω−k3−r2 , ω−k4−r4 ; r1 − r3) +O

(1

T+

1

λ+ `λ,a,n

),

and

Tλd

2cov

[Ag,h(r1, r2), Ag,h(r3, r4)

]=

2

Tλd

T/2∑k2,k4=1

h(ωk2)h(ωk4)b(2)r2,r4,k2,k4

(ωk2 , ωk4 ; r1, r3)

+2

T 2λ2d

T/2∑k2,k4=1

a∑k1,k3=−a

h(ωk2)h(ωk4)g(Ωk1)g(Ωk3)

×br2+r4,4(Ωk1+r1 , ωk2+r2 ,Ω−k3 , ω−k3 ,Ωk3+r3 , ωk4+r4 ; r1 + r3) +O

(1

T+

1

λ+ `λ,a,n

).

7 Simulations

7.1 Set-up

We now assess the finite sample performances of the test statistics described above through

simulations. In all cases we consider mean zero spatio-temporal processes, where T = 200

and at each time point we observe n = 100 or 500 locations (the locations are drawn from

a uniform distribution defined on [−λ/2, λ/2]2 and we use the same set of locations at each

time point). All tests are done at the 5% level and all results are based on 300 replications.

Further, we investigate the performances of the tests when the coefficients ag(ωk; r1, r2), as

defined in Section 4.1, are being calculated while both removing and keeping the ‘nugget

effect’ NT (in Tables 1-5 the rejection rates for the test statistics without removing NT are

reported in the parentheses). All simulations are done for spatial dimension d = 2.

Next, we briefly discuss the implementation issues.

1. Choice of set P and P ′: All test statistics depend on the choice of P and P ′. In

all simulations described in this section we use P = (1, 0), (1, 1), (0, 1), (−1, 1) ×

39

1, 2 and P ′ = (1, 0), (1, 1), (0, 1), (−1, 1) × 4, 5 to calculate empirical type I

errors and overall powers. Further, to test for stationarity over space, we take P =

(1, 0), (1, 1), (0, 1), (−1, 1) × 0 and P ′ = (2, 0), (2, 1), (2, 2), (1, 2), (0, 2), (−1, 2),

(−2, 2), (−2, 1) × 0 and to test for stationarity over time, we take P = (0, 0) ×1, 2 and P ′ = (0, 0) × 4, 5.

2. Choice of g(·) : Based on the discussion in Remark 2.1 we use the weight function

g(Ω) =∑L

j=1 eiv′jΩ. The choice of vj’s should depend on the density of the sampling

region. Following the same rationale as described in Bandyopadhyay and Subba Rao

[2016], in all simulations we define the v grid as V = vj = (vj1, vj2)′ ∈ Rd : vjk =

−s,−s/2, 0, s/2, s, for k = 1, 2 such that vj+vj′ 6= 0 for vj,vj′ ∈ V, where s = λ/n1/d

is the ‘average spacing’ between the observations on each axis. We should mention

that if the support of the empirical covariance of the data appears far greater than

s = λ/n1/d, then using a wider v grid is appropriate. If changes in the spatial covariance

function happen mainly at lags much smaller than s = λ/n1/d, then data is not available

to detect changes in the spatial covariance structure.

3. Choice of frequency grid : In all simulations we use a =√n in the definition of

ag(ωk; r1, r2).

4. Choice of H in the definition of T2 and M2: For all simulations we use H = 10 and

H = 20.

5. Choice of M to calculate the local averages : In order to estimate Vg(ωk) we use the

estimator Vg(ωk;P ′) (defined in (28)) with M = 2 (thus taking a local average of 5).

To obtain the critical values of the tests we use the asymptotic ‘finite sample’ approx-

imations of the distributions of the test statistics as described in Section 4.4. For ease

of discussion below we refer to (i) T1,g,V −1/2 and M1,g,V −1/2 as the average covariance test

statistics (ii) T2,g,V −1/2,1 and M2,g,V −1/2,1 as the average squared covariance test statistics and

(iii) T2,g,V −1/2,W and M2,g,V −1/2,W as the variance adjusted average squared covariance test

statistics.

7.2 Simulations under the null

7.2.1 Models

In order to define the spatio-temporal models, we start by defining the ‘innovations’ process.

Let εt(s); s ∈ R2, t ∈ Z denote a spatio-temporal stationary Gaussian random field which

is independent over time with spatial exponential covariance cov[εt(s1), εt(s2)] = κ0(‖s2 −s1‖2) = exp(−‖s1 − s2‖2/ρ), where ρ is the ‘range parameter’. We do all the simulations

40

under the null with ρ = 0.5, ρ = 1 and λ = 5 (in the case that ρ = 1 the range of dependence

for the innovations is 20%, whereas for ρ = 0.5 the range reduces to 10%; see Figure 1). We

mention that the decorrelation property of Fourier Transforms, given in Lemma 3.1 implicitly

depends on the range of dependence with respect to λ. If the range of dependence is too

large with respect to the observed random field then the degree of correlation in the Fourier

transforms will be non-negligible (leading to false rejection of the null).

(S1) Spatially and temporally stationary Gaussian random field : We define a spatio-temporal

model with the temporal AR(1) structure Zt(s) = 0.5Zt−1(s) + εt(s).

(S2) Spatially and temporally stationary non-Gaussian random field: To induce non-linearity

and non-Gaussianity in the random field we use a Bilinear model of the form

Zt(s) = 0.5Zt−1(s) + 0.4Zt−1(s)εt−1(s) + εt(s).

We note that the nonlinear term 0.4Zt−1(s)εt−1(s) induces sporadic bursts in the

spatio-temporal process. The coefficients 0.5 and 0.4 are chosen to ensure that the

process has a finite second moment (see Subba Rao and Gabr [1984] for details).

7.2.2 Discussion

The results for model S1 and S2 are given Table 1.

We first consider the stationary Gaussian model (S1). The results for all the tests are

relatively good for both ρ = 0.5 and ρ = 1. However, for the average squared statistics

(without variance adjustment) for H = 10 and ρ = 1 there are some inflations in the type

I error. This is probably because without the variance adjustment the average squared

statistics depend on the asymptotic result Wg,V −1/2(ωjH)P→ 1 (see (36)) which depends on

the range parameter ρ and the degree of non-Gaussianity. Model S1 is Gaussian, and it

seems the error in this approximation seems only to mildly impact the case H = 10 and

ρ = 1.

The results from the simulations for the stationary but non-Gaussian model (S2) are very

different. The average covariance test results keep close to the nominal level (for both ρ = 0.5

and 1) however there is a substantial inflation in the type I error (between 70-90%) for the

average squared statistic without variance adjustment (for both H = 10 and H = 20). This

is likely due to the non-Gaussianity of the process which seems to greatly impact the rate

that Wg,V −1/2(ωjH)P→ 1. However, the variance adjusted average squared covariance test

statistics appear to keep close to the nominal level for both ρ = 0.5 and 1 and H = 10 and

20. This demonstrates that Wg,V −1/2(ωjH)P→ 1 is an asymptotic result and for finite samples

it is important to estimate the variance.

41

In all cases, both removing and keeping the nugget term NT give comparable results.

Our results in the simulation study demonstrate that both the average covariance and

the variance adjusted average squared covariance test statistics perform well under the null,

but caution needs to be taken when interpreting the results of the non-variance adjusted

average squared covariance tests.

Model S1 Model S2ρ ρ

0.5 1 0.5 1n 100 500 100 500 100 500 100 500

T1,g,V −1/2 0.08 (0.08) 0.08 (0.07) 0.07 (0.07) 0.04 (0.04) 0.09 (0.09) 0.08 (0.09) 0.09 (0.08) 0.08 (0.07)

M1,g,V −1/2 0.04 (0.07) 0.05 (0.06) 0.05 (0.06) 0.02 (0.02) 0.06 (0.04) 0.05 (0.06) 0.07 (0.07) 0.06 (0.08)

H=20

T2,g,V −1/2,1 0.04 (0.03) 0.01 (0.01) 0.06 (0.04) 0.03 (0.02) 0.45 (0.70) 0.86 (0.91) 0.62 (0.72) 0.94 (0.98)

M2,g,V −1/2,1 0.07 (0.08) 0.06 (0.05) 0.07 (0.08) 0.02 (0.03) 0.48 (0.67) 0.80 (0.88) 0.60 (0.70) 0.86 (0.88)

T2,g,V −1/2,W 0.05 (0.05) 0.02 (0.01) 0.07 (0.03) 0.02 (0.01) 0.06 (0.07) 0.05 (0.04) 0.05 (0.05) 0.06 (0.05)

M2,g,V −1/2,W 0.05 (0.06) 0.03 (0.03) 0.05 (0.05) 0.06 (0.05) 0.04 (0.07) 0.06 (0.07) 0.10 (0.09) 0.08 (0.08)

H=10

T2,g,V −1/2,1 0.10 (0.10) 0.08 (0.07) 0.12 (0.12) 0.13 (0.14) 0.50 (0.71) 0.85 (0.90) 0.67 (0.78) 0.85 (0.88)

M2,g,V −1/2,1 0.08 (0.08) 0.09 (0.08) 0.11 (0.12) 0.16 (0.15) 0.42 (0.58) 0.65 (0.77) 0.53 (0.67) 0.79 (0.83)

T2,g,V −1/2,W 0.10 (0.08) 0.04 (0.04) 0.06 (0.04) 0.04 (0.04) 0.09 (0.10) 0.06 (0.06) 0.05 (0.06) 0.08 (0.10)

M2,g,V −1/2,W 0.05 (0.06) 0.03 (0.04) 0.06 (0.05) 0.05 (0.05) 0.08 (0.10) 0.09 (0.09) 0.10 (0.13) 0.11 (0.12)

Table 1: Empirical type I errors at 5% level based on different tests with λ = 5 for Gaussianand non-Gaussian stationary data with innovations coming from a Gaussian random fieldwith exponential covariance functions. Rejection rate without removing NT (see (18)) are inthe parentheses.

7.3 Simulations under the alternative

7.3.1 Models

In order to induce spatial nonstationarity in the models (NS2) and (NS3) (defined below) we

define the Gaussian innovations process ηt(s); s ∈ [−λ/2, λ/2]2, which is independent over

time with nonstationary covariance cov[ηt(s1), ηt(s2)] = cλ(s1, s2) = κ0(s2 − s1; s1) where,

cλ(s1, s2) = |Σ(s1

λ

)|1/4|Σ

(s2

λ

)|1/4∣∣∣∣Σ(s1

λ) + Σ(s2

λ)

2

∣∣∣∣−1/2

exp[−√Qλ(s1, s2)],

| · | denotes the determinant of a matrix, Qλ(s1, s2) = 2(s1 − s2)′[Σ(s1

λ) + Σ(s2

λ)]−1(s1 − s2)

and Σ(sλ) = Γ(s

λ)ΛΓ(s

λ)′, where

Γ(sλ

)=

[γ1(s/λ) −γ2(s/λ)

γ2(s/λ) γ1(s/λ)

], Λ =

[1 0

0 12

],

with γ1(s/λ) = log (sx/λ+ 0.75), γ2(s/λ) = (sx/λ)2+(sy/λ)2, and s = (sx, sy)′(see Paciorek

and Schervish [2006] and Jun and Genton [2012] for the details on this process). Note that

42

the variance of this process is constant over the spatial random field and it is simply the

correlation structure that varies over space.

(NS1) Temporally nonstationary but spatially stationary Gaussian random field: Zt(s) =

0.5Zt−1(s) +(1.3 + sin

(2πt400

))εt(s), where εt(s) is defined in Section 7.2.1. We use

ρ = 0.5, ρ = 1 and λ = 5.

(NS2) Temporally stationary but spatially nonstationary Gaussian random field: The spatio-

temporal process is defined with an AR(1) model Zt(s) = 0.5Zt−1(s)+ηt(s). Following

a similar set-up as in Bandyopadhyay and Subba Rao [2016] we use λ = 20. This

process has a constant variance over space and time.

(NS3) Both temporally and spatially nonstationary Gaussian random field: The spatio-temporal

process is defined using an AR(1) model with time-dependent innovations

Zt(s) = 0.5Zt−1(s) +

(1.3 + sin

(2πt

400

))ηt(s).

For the simulations we use λ = 20.

7.3.2 Discussion

The empirical powers based on Models NS1 - NS3 are given in Table 2-5.

First we consider Model NS1, which is temporarily nonstationary, but stationary over

space. The results of the general spatio-temporal test using the test set P = (1, 0), (1, 1),

(0, 1), (−1, 1)×1, 2 and orthogonal estimates set P ′ = (1, 0), (1, 1), (0, 1), (−1, 1)×4, 5(described in Section 4) are given in Table 2. Before discussing the results we note that

over the test set P the Fourier transforms are near uncorrelated. However, the temporal

nonstationarity means that the orthogonal estimators Ag,h(r1, r2) and Bg,h;H(ωjH ; r1, r2) for

(r1, r2) ∈ P ′ do not necessarily share the same variance. Furthermore, there is correlation

between the terms. These conflicting behaves (decorrelation of DFTs but inability to capture

the true variance) helps explain why the power in the overall test varies between 27%-80%

in the case ρ = 0.5 and 21% - 80% in the case ρ = 1 (excluding the non-variance adjusted

tests). The results of the one-way temporal stationary and one-way spatial stationary tests

(described in Section 5) are given in Table 3. The power in the one-way temporal tests are

close to 100% for all the test statistics (as we would expect since the process is temporally

nonstationary) for both ρ = 0.5 and ρ = 1. The power for the one-way spatial tests drops

considerably (as expected because NS1 is spatially stationary) for the average covariance test

and variance adjusted average squared covariance test. In the case of the variance adjusted

average squared tests the proportion of rejection is least in the case ρ = 0.5 and H = 10.

43

Next we consider Model NS2, which is temporarily stationary, but spatially nonstationary.

The results are reported in Table 4. In this case the general spatio-temporal test using

the test set P = (1, 0), (1, 1), (0, 1), (−1, 1) × 1, 2 and orthogonal estimates set P ′ =

(1, 0), (1, 1), (0, 1), (−1, 1)×4, 5 gives very little power. As we would expect, in the one-

way test for temporal stationarity the proportion of rejections is close to the nominal level

(with the exception of the variance adjusted test average squared test with H = 20 when

the proportion of rejection is about 12%). However, the test does seem to have some power

in the one-way test for spatial stationarity. In the case that n = 500 all the tests (excluding

the non-variance adjusted tests) have power between 8-21%. This level of power is not high

but it is higher than the case n = 100. The overall low power is because the number of

observations is relatively sparse on the random field (n = 500 and λ = 20). Therefore

most of the observations are unlikely to be highly correlated and thus contains very little

information about the nonstationary structure (recall the variance of the spatio-temporal

process is constant). It is likely if a larger n were used in the simulations, the power would

increase (compare with the simulations in Bandyopadhyay and Subba Rao [2016]).

Lastly, we consider Model NS3, which is both temporal and spatial nonstationarity. The

results are presented in Table 5. For the general spatio-temporal tests we get higher powers

than for Model NS1 across all the tests. The power increases to 100% for the one-way

temporal stationary test. For the one-way spatial stationarity tests the power is more than

for the same tests using model NS2.

We mention that for all the models (NS1-NS3) the power for the average squared covari-

ance test without variance adjustment is very high. However, we have to be cautious about

interpreting the result of these tests as the simulations under the null of stationarity show

that the these test statistics are unable to keep the nominal level when the process is not

Gaussian.

Comparing the rejection rates with and without the nugget term removed (the values

outside and insides the parentheses), we observe that for models NS1 and NS2 the rejection

rates with and without the nugget term are about the same. However, for NS3 the power is

slightly more after removing the nugget term.

Dedication

SSR was very fortunate to attend a course on nonparametric statistics given by Professor

M. B. Priestley when she was an undergraduate student. His classes were a joy to attend.

During the 1960’s, Professor M. B. Priestley was one the first researchers to study non-

stationary time series, without his fundamental contributions this paper would not have

been possible. Therefore, this paper is dedicated to the memory of Professor M. B. Priestley

whose kind nature and encouragement was an inspiration to all.

44

Model NS1: Overall Powerρ

0.5 1n 100 500 100 500

T1,g,V −1/2 0.73 (0.80) 0.60 (0.59) 0.76 (0.74) 0.57 (0.49)

M1,g,V −1/2 0.74 (0.78) 0.61 (0.59) 0.80 (0.75) 0.59 (0.51)

H=20

T2,g,V −1/2,1 0.99 (0.99) 0.99 (0.99) 0.99 (1.00) 0.97 (0.96)

M2,g,V −1/2,1 0.97 (0.99) 0.99 (0.99) 0.98 (0.98) 0.93 (0.92)

T2,g,V −1/2,W 0.44 (0.56) 0.33 (0.27) 0.45 (0.45) 0.22 (0.21)

M2,g,V −1/2,W 0.51 (0.64) 0.47 (0.45) 0.60 (0.57) 0.34 (0.27)

H=10

T2,g,V −1/2,1 1.00 (1.00) 1.00 (0.99) 0.99 (1.00) 1.00 (1.00)

M2,g,V −1/2,1 0.98 (0.99) 0.99 (0.98) 0.99 (0.98) 0.98 (0.97)

T2,g,V −1/2,W 0.53 (0.56) 0.39 (0.37) 0.55 (0.50) 0.30 (0.27)

M2,g,V −1/2,W 0.52 (0.56) 0.46 (0.44) 0.52 (0.48) 0.33 (0.28)

Table 2: Overall empirical power at 5% level based on different tests with λ = 5 for non-stationary data generated from the model NS1 with innovations coming from a Gaussianrandom field with exponential covariance functions. Rejection rate without removing NT

(see (18)) are in the parentheses.

Model NS1Temporal Power Spatial Power

ρ ρ0.5 1 0.5 1

n 100 500 100 500 100 500 100 500T1,g,V −1/2 0.97 (0.99) 1.00 (1.00) 0.97 (0.99) 1.00 (1.00) 0.01 (0.01) 0.01 (0.01) 0.04 (0.03) 0.04 (0.04)

M1,g,V −1/2 0.99 (1.00) 1.00 (1.00) 0.99 (0.99) 1.00 (1.00) 0.01 (0.01) 0.02 (0.01) 0.02 (0.02) 0.03 (0.01)

H=20

T2,g,V −1/2,1 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (0.99) 0.99 (0.99) 1.00 (1.00) 1.00 (1.00)

M2,g,V −1/2,1 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (0.98) 0.99 (0.99) 1.00 (1.00) 1.00 (1.00)

T2,g,V −1/2,W 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.05 (0.04) 0.10 (0.08) 0.15 (0.14) 0.28 (0.31)

M2,g,V −1/2,W 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.15 (0.15) 0.18 (0.16) 0.23 (0.24) 0.31 (0.32)

H=10

T2,g,V −1/2,1 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (0.99) 0.99 (0.99) 1.00 (1.00) 1.00 (1.00)

M2,g,V −1/2,1 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (0.99) 0.99 (0.99) 1.00 (1.00) 1.00 (1.00)

T2,g,V −1/2,W 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.02 (0.02) 0.06 (0.05) 0.10 (0.11) 0.26 (0.28)

M2,g,V −1/2,W 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.06 (0.06) 0.09 (0.07) 0.14 (0.14) 0.24 (0.26)

Table 3: One-way empirical powers at 5% level based on different tests with λ = 5 fornonstationary data generated from the model NS1 with innovations coming from a Gaussianrandom field with exponential covariance functions. Rejection rate without removing NT

(see (18)) are in the parentheses.

Acknowledgments

SB’s and SSR’s work has been partially supported by the National Science Foundation, DMS-

1406622 and DMS-1513647 respectively. CJ has been supported by the Research Center 884

“Political Economy of Reforms” (Project B6) funded by the German Science Foundation

(DFG). The authors also wish to thank Tilmann Gneiting and Tata Subba Rao who encour-

aged us to consider this problem.

45

Model NS2Overall Power Temporal Power Spatial Power

n 100 500 100 500 100 500T1,g,V −1/2 0.09 (0.11) 0.11 (0.11) 0.04 (0.05) 0.05 (0.06) 0.06 (0.07) 0.17 (0.17)

M1,g,V −1/2 0.08 (0.08) 0.10 (0.09) 0.03 (0.03) 0.05 (0.06) 0.03 (0.02) 0.15 (0.15)

H=20

T2,g,V −1/2,1 0.04 (0.06) 0.06 (0.06) 0.04 (0.05) 0.05 (0.06) 0.24 (0.26) 0.28 (0.31)

M2,g,V −1/2,1 0.07 (0.09) 0.07 (0.07) 0.08 (0.08) 0.03 (0.04) 0.12 (0.15) 0.21 (0.25)

T2,g,V −1/2,W 0.07 (0.06) 0.05 (0.04) 0.11 (0.12) 0.12 (0.12) 0.05 (0.06) 0.18 (0.19)

M2,g,V −1/2,W 0.08 (0.06) 0.08 (0.09) 0.11 (0.12) 0.13 (0.12) 0.07 (0.10) 0.21 (0.20)

H=10

T2,g,V −1/2,1 0.15 (0.18) 0.15 (0.15) 0.06 (0.07) 0.06 (0.07) 0.38 (0.47) 0.56 (0.59)

M2,g,V −1/2,1 0.12 (0.10) 0.13 (0.15) 0.05 (0.06) 0.06 (0.06) 0.32 (0.34) 0.48 (0.50)

T2,g,V −1/2,W 0.08 (0.10) 0.06 (0.05) 0.05 (0.04) 0.04 (0.04) 0.01 (0.02) 0.09 (0.10)

M2,g,V −1/2,W 0.13 (0.14) 0.10 (0.10) 0.06 (0.03) 0.04 (0.05) 0.01 (0.01) 0.08 (0.09)

Table 4: Empirical powers at 5% level based on different tests with λ = 20 for nonstationarydata generated from the model NS2. Rejection rate without removing NT (see (18)) are inthe parentheses.

Model NS3Overall Power Temporal Power Spatial Power

n 100 500 100 500 100 500T1,g,V −1/2 0.83 (0.80) 0.98 (0.92) 0.92 (0.99) 1.00 (1.00) 0.11 (0.07) 0.33 (0.19)

M1,g,V −1/2 0.92 (0.88) 0.99 (0.97) 0.95 (1.00) 1.00 (1.00) 0.18 (0.08) 0.54 (0.25)

H=20

T2,g,V −1/2,1 0.99 (0.99) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 0.99 (1.00) 1.00 (1.00)

M2,g,V −1/2,1 0.99 (0.98) 1.00 (0.99) 1.00 (1.00) 1.00 (1.00) 0.99 (1.00) 1.00 (1.00)

T2,g,V −1/2,W 0.65 (0.66) 0.85 (0.77) 1.00 (1.00) 1.00 (1.00) 0.34 (0.22) 0.74 (0.50)

M2,g,V −1/2,W 0.82 (0.80) 0.98 (0.87) 1.00 (1.00) 1.00 (1.00) 0.52 (0.34) 0.90 (0.69)

H=10

T2,g,V −1/2,1 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00)

M2,g,V −1/2,1 0.99 (0.99) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00) 1.00 (1.00)

T2,g,V −1/2,W 0.65 (0.63) 0.81 (0.77) 1.00 (1.00) 1.00 (1.00) 0.13 (0.08) 0.45 (0.35)

M2,g,V −1/2,W 0.79 (0.70) 0.94 (0.84) 1.00 (1.00) 1.00 (1.00) 0.32 (0.19) 0.79 (0.58)

Table 5: Empirical powers at 5% level based on different tests with λ = 20 for nonstationarydata generated from the model NS3. Rejection rate without removing NT (see (18)) are inthe parentheses.

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49

A Supplementary material

A.1 Proof of Lemma 3.1

To prove the result we start by expanding cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)].

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)]

=1

2π

T−1∑h=−(T−1)

e−ihωk21

T

T−max(0,h)∑t=1−min(0,h)

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]e−itωr2

= M +R, (48)

where M is the main term

M =1

2π

T−1∑h=−(T−1)

e−ihωk21

T

T∑t=1

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]e−itωr2 ,

and R is the remainder

R =1

2π

T−1∑h=0

e−ihωk21

T

T∑t=T−h+1

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]e−itωr2

+1

2π

−1∑h=−(T−1)

e−ihωk21

T

|h|∑t=1

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]e−itωr2 .

The expansions above are valid in the general case. Below we obtain expressions for M (the

main term) and bounds for R in the case that the spatio-temporal process is stationary and

nonstationary.

• Spatially stationary

By using the same proof used to prove Theorem 2.1(i), Bandyopadhyay and Subba Rao

[2016], and the rescaling devise over time, under spatial stationary we have, for r1 6= 0,

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]

=

∫[−λ/2,λ/2]d

e−iΩ′rs2

∫[−λ/2−s1,−λ/2]d

κh; tT

(s1)eiΩ′k1

s1ds1ds2︸ ︷︷ ︸O(

ρhλd−b

)

+

∫[−λ/2,λ/2]d

e−iΩ′rs2

∫[λ/2,λ/2+s1]d

κh; tT

(s1)eiΩ′k1

s1ds1ds2︸ ︷︷ ︸O(

ρhλd−b

)

+O( ρhTλd−b

ITime=NS

),

50

and for r1 = 0,

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]

=ct,t+hλd

∫[−λ/2,λ/2]d

κh; tT

(v) exp(−iv′Ωk1)dv +λdnt,t+hntnt+h

κh, tT

(0) +O

(ρhITime=NS

T

),

where, b = b(r1) is the number of zeros in r1, ct,t+h = (ntnt+h − nt,t+h)/ntnt+h,

nt,t+h = |st,jntj=1 ∩ st+h,jnt+hj=1 | and ITime=NS denotes the indicator variable for tem-

poral nonstationarity. Note that we use the notation [−λ/2 − s1,−λ/2]d = [−λ/2 −s11] × . . . × [−λ/2 − s1d]. Substituting the above into the remainder R we see that

|R| = O([T−1 +λd/n]I(r1 = 0) + 1λd−bT

I(r1 6= 0)). Now we derive expression for M for

the temporally stationary and nonstationary separately.

(a) Temporally stationary (i.e., κh, tT

(v) = κh(v)) First we look at the case r1 6=0. In the case that r1 6= 0 and r2 6= 0, we take the summand

∑Tt=1 e

−itωr2 in

M separate of κh giving M = 0. Therefore, cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)] =

O(λ−(d−b)T−1). In the case that r1 6= 0 but r2 = 0, we get M = O(λ−(b−d)), and thus

cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)] = O(λ−(d−b)).

Now we consider the case r1 = 0. In the case that r1 = 0 but r2 6= 0, we use

Assumption 3.1(ii), where c1n ≤ nt ≤ c2n, which implies that |ct,t+h − 1| ≤ c2c1n

and immediately gives M = O(T−1 + λd/n) and cov [J(Ωk1 , ωk2), J(Ωk1 , ωk2+r2)] =

O(λd/n+T−1). On the other hand, when r1 = 0 and r2 = 0 we have M = f(Ωk1 , ωk2)+

O(T−1 + λ−1 + λd/n), which immediately leads us to cov [J(Ωk1 , ωk2), J(Ωk1 , ωk2)] =

f(Ωk1 , ωk2) +O(T−1 + λ−1 + λd/n).

(b) Temporally nonstationary Again it is immediately clear that when r1 6= 0

(r2 ∈ Z) we have M = O(λ−(d−b)), which gives cov [J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2)] =

O(λ−(d−b) + T−1). However, when r1 = 0 (r2 ∈ Z) (and using Assumption 3.1(ii)) it is

clear that

M =1

2π

T−1∑h=−(T−1)

e−ihωk21

T

T∑t=1

1

λd

∫[−λ/2,λ/2]d

κh; tT

(v) exp(−iv′Ωk1)dv +O

(λd

n+

1

T

),

which gives the desired result.

• Spatially nonstationary If the spatio-temporal process is spatially nonstationary,

using the same proof to prove Theorem 2.1(ii), Bandyopadhyay and Subba Rao [2016]

51

and the rescaling devise over time and space we have,

cov[Jt(Ωk1), Jt+h(Ωk1+r1)]

=ct,t+hλd

∫[−λ/2,λ/2]2d

κh; tT

(v;

s

λ

)exp(−iv′Ωk1) exp(−is′Ωr1)dvds

+

∫[−λ/2,λ/2]d

e−iΩ′rs

∫[−λ/2−s,−λ/2]d

κh; tT

(v;

s

λ

)eiΩ

′k1

vdvds︸ ︷︷ ︸O(ρh/λ)

+

∫[−λ/2,λ/2]d

e−iΩ′rs

∫[λ/2,λ/2+s]d

κh; tT

(v;

s

λ

)eiΩ

′k1

vdvds︸ ︷︷ ︸O(ρh/λ)

+nt,t+hntnt+h

∫[−λ/2,λ/2]d

κh; tT

(0;

s

λ

)exp(−is′Ωr1)ds +O

(ρhITime=NS

T

).

Using the above result it is straightforward to show that R = O([1 + λd/n]T−1).

(a) Temporally stationary (i.e., κh, tT

(v, s) = κh(v, s)). Since the process is spatially

nonstationary, we consider r1 = 0 and r1 6= 0 together. In the case that r2 6= 0∑Tt=1 e

−itr2 is separate of κh, thus M = 0 and cov [J(Ωk1 , ωk2), J(Ωk1 , ωk2)] = O(T−1).

If r2 = 0 we have,

M =1

2π

T−1∑h=−(T−1)

e−ihωk21

λd

∫[−λ/2,λ/2]d

κh

(v;

s

λ

)exp(−iv′Ωk1) exp(−is′Ωr1)dvds+O

(1

λ

),

which immediately leads to the desired result.

(b) Temporally nonstationary In this case using Assumption 3.1(ii) we have,

M =1

2π

T−1∑h=−(T−1)

e−ihωk21

T

T∑t=1

e−itωr2

×ct,t+hλd

∫[−λ/2,λ/2]2d

κh; tT

(v;

s

λ

)exp(−iv′Ωk1) exp(−is′Ωr1)dvds +O

(λd

n+

1

λ

),

thus leading to the desired result.

A.2 Proof of results for stationary spatio-temporal processes

PROOF of Lemma 4.1 The proof of this lemma is identical to the proof of Lemma 3.1 in

Bandyopadhyay and Subba Rao [2016] and hence omitted.

52

To prove the remainder of the results in Section 4 we use the following notation

fh(Ω) =

∫Rdκh(s) exp(−iΩ′s)ds,

f(Ω, ω) =∑h∈Z

exp(−ihω)

∫Rdκh(s) exp(−iΩ′s)ds,

and

fh1,h2,h3(Ω1,Ω2,Ω3) =

∫R3d

κh1,h2,h3(s1, s2, s3) exp(−i(s′1Ω1 + s′2Ω2 + s′3Ω3))ds1ds2ds3.

Note that in this section we do not prove any central limit theorems. However, we conjecture

that by combining Bandyopadhyay et al. [2015], which give a CLT for mixing spatial processes

and the CLT for quadratic forms of a time series (see, for example, Hsing and Wu [2004],

Leucht [2012], Lee and Subba Rao [2015]) asymptotic normality of spatio-temporal quadratic

forms can be proved.

Having established an expression for the mean of ag(·) under stationarity, the main focus

is obtaining expressions for the variance and covariance of ag(·) and the corresponding test

statistics. To do this we define the related quantity ag(·) such that

ag(ωk2 ; r1, r2) = ag(ωk2 ; r1, r2) +NT ,

where,

NT =1

2πT

T∑t,τ=1

eitωk2−iτωk2+r21

λd

a∑k1=−a

g(Ωk1)1

ntnτ

n∑j=1

δt,jδτ,jZt(sj)Zτ (sj)e−isjΩr1 .

More precisely, we have,

ag(ωk2 ; r1, r2)

=1

λd

a∑k1=−a

g(Ωk1)J(Ωk1 , ωk2)J(Ωk1+r1 , ωk2+r2)

=1

2πT

T∑t,τ=1

eitωk2−iτωk2+r21

λd

a∑k1=−a

g(Ωk1)Jt(Ωk1)Jτ (Ωk1+r1)

=1

2πT

T∑t,τ=1

eitωk2−iτωk2+r21

λd

a∑k1=−a

g(Ωk1)1

ntnτ

n∑j1,j2=1

δt,j1δτ,j2Zt(sj1)Zτ (sj2)

×eisj1Ωk1−isj2Ωk1+r1 ,

53

where the second equation follows by expanding J(Ωk1 , ωk2). To understand the role NT

plays, consider the expectation of NT for the case r1 = 0 and r2 = 0; not a case included in

the text, but useful in understanding its role. Taking expectation of NT (under stationarity)

we have

E[NT ] =1

2πT

a∑k=−a

g(Ωk1)T∑

t,τ=1

exp(−i(τ − t)ωk2)nt,τntnτ

≈ λd

n

∫2π[−a/λ,a/λ]d

g(Ω)dΩ× 1

2π

∑h∈Z

exp(−ihωk2)κh(0) = O

(ad

n

).

In the case that we constrain the frequency grid Ωk;k = (k1, . . . , kd),−a ≤ kj ≤ a to

be bounded, i.e., a/λ → c < ∞ as a, λ → ∞, then it is clear that E[NT ] = O(λd/n) =

o(1). Furthermore, using similar arguments it can be shown that the variance of NT is

asymptotically negligible and λdvar[ag(ωk2 ; r1, r2)] = λdvar[ag(ωk2 ; r1, r2)] + o(1) when the

frequency grid is bounded. On the other hand, if the frequency grid is not bounded and

a/λ → ∞ as λ → ∞ then we can show that for r1 = 0 and r2 = 0 we have E[NT ] =

O(ad/n) and for general r1 and r2 λdvar[NT ] = (a2d/n2). Therefore, if the frequency grid is

not bounded, ag(ωk2 ; r1, r2) and ag(ωk2 ; r1, r2) are not asymptotically equivalent. However,

ag(ωk2 ; r1, r2) does play an important role in understanding the covariance of ag(ωk2 ; r1, r2),

and we come back to this later on.

Returning to ag(ωk2 ; r1, r2), we see from the definition of ag(·) that in order to obtain the

covariance of ag(·) we require the expansion

λdcov

[1

λd

a∑k1=−a

g(Ωk1)1

nt1nt2

∑j1 6=j2

δt1,j1δt2,j2Zt1(sj1)Zt2(sj2)eisj1Ωk1

−isj2Ωk1+r1 ,

1

λd

a∑k3=−a

g(Ωk3)1

nt3nt4

∑j3 6=j4

δt3,j3δt4,j4Zt4(sj3)Zt4(sj4)eisj3Ωk3

−isj4Ωk3+r3

]= A + B + C (49)

where,

A =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)1∏4

j=1 ntj

∑j1 6=j2j3 6=j4

cov[δt1,j1Zt1(sj1)e

isj1Ωk1 , δt3,j1Zt3(sj1)eisj3Ωk3

]×cov

[δt2,j2Zt2(sj2)e

−isj2Ωk1+r1 , δt4,j4Zt4(sj4)e−isj4Ωk3+r3

]

54

B =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)1∏4

j=1 nj

∑j1 6=j2j3 6=j4

cov[δt1,j1Zt1(sj1)e

isj1Ωk1 , δt4,j4Zt4(sj4)e−isj4Ωk3+r3

]×cov

[δt2,j2Zt2(sj2)e

−isj2Ωk1+r1 , δt3,j1Zt3(sj1)eisj3Ωk3

]

C =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)1∏4

j=1 nj

∑j1 6=j2j3 6=j4

cum

[δt1,j1Zt1(sj1)e

isj1Ωk1 , δt2,j2Zt2(sj2)e−isj2Ωk1+r1 ,

δt3,j1Zt3(sj1)e−isj3Ωk3 , δt4,j4Zt4(sj4)e

isj4Ωk3+r3

].

Simplifications for these terms can be obtained by using the methods developed in Subba Rao

[2015a]. Using this we can show

A =Ir1=r3

(2π)d

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr1)dΩ +R1,t3−t1,t4−t2 ,

B =Ir1=r3

(2π)d

∫Dr1

g(Ω)g(−Ω−Ωr1)ft4−t1(Ω)ft3−t2(Ω + Ωr)dΩ +R2,t4−t1,t3−t2

and,

C =Ir1=r3

(2π)2d

∫D2

g(Ω1)g(Ω2)ft2−t1,t3−t1,t4−t1(Ω1 + Ωr1 ,Ω2,−Ω2 −Ωr1)dΩ1dΩ2

+R3,t2−t1,t3−t1,t4−t1 ,

where,

|R1,t3−t1,t4−t2| = O(ρt3−t1ρt4−t2`λ,a,n),

|R2,t4−t1,t3−t2| = O(ρt4−t1ρt3−t2`λ,a,n), and

R3,t2−t1,t3−t1,t4−t1 = O

(ρt2−t1ρt3−t1ρt4−t1

[`λ,a,n +

(aλ)d

n2

]).

We further observe that use of the expansions given in (49) to obtain an expression

for var[ag(ωk; r1, r2)] can make the notations extremely cumbersome and difficult to follow.

Proofs which only involve DFTs can substantially reduce cumbersome notations. However,

a DFT based proof requires the frequency grid to be bounded, and as mentioned in the

discussion at the start of this section, ag(ωk2 ; r1, r2) and ag(ωk2 ; r1, r2) are only asymptotically

equivalent if the frequency grid is bounded. Therefore to simplify notations, for the remainder

of this section we focus on the case that the frequency grid is bounded. However, we mention

55

that exactly the same bounds apply to the case when the frequency grid is unbounded.

We observe that in order to obtain an expression for λdcov[ag(ωk1 ; r1, r2), ag(ωk2 ; r3, r4)]

(in the case that the frequency grid is bounded) we require the expansion

λdcov

[a∑

k1=−a

g(Ωk1)Jt1(Ωk1)Jt2(Ωk1+r1),a∑

k3=−a

g(Ωk3)Jt2(Ωk3)Jt4(Ωk3+r3)

]= A + B + C,

where

A =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cov [Jt1(Ωk1), Jt3(Ωk3)] cov[Jt2(Ωk1+r1), Jt4(Ωk3+r3)

],

B =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cov[Jt1(Ωk1), Jt4(Ωk3+r3)

]cov

[Jt2(Ωk1+r1), Jt3(Ωk3)

],

C =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cum[Jt1(Ωk1), Jt2(Ωk1+r1), Jt3(Ωk3), Jt4(Ωk3+r3)

].

Now we obtain simplified expressions for A, B and C.

A =Ir1=r3

(2π)d

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr1)dΩ +R1,t3−t1,t4−t2 , (50)

B =Ir1=r3

(2π)d

∫Dr1

g(Ω)g(−Ω−Ωr1)ft4−t1(Ω)ft3−t2(Ω + Ωr)dΩ +R2,t4−t1,t3−t2 , (51)

C =Ir1=r3

(2π)2d

∫D2

g(Ω1)g(Ω2)ft2−t1,t3−t1,t4−t1(Ω1 + Ωr1 ,Ω2,−Ω2 −Ωr1)dΩ1dΩ2

+R3,t2−t1,t3−t1,t4−t1 . (52)

Comparing the above with (49), when the frequency grid is unbounded, see that the expres-

sions are identical. We use the above to prove Lemma 4.2.

PROOF of Lemma 4.2 By decomposing the covariance we have

λdcov [ag(ωk2 ; r1, r2), ag(ωk4 ; r3, r4)] = Ik2,k4 + IIk2,k4 + IIIk2,k4 ,

56

where,

Ik2,k4 =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cov [J(Ωk1 , ωk2), J(Ωk3 , ωk4)]

×cov[J(Ωk1+r1 , ωk2+r2), J(Ωk3+r3 , ωk4+r4)

],

IIk2,k4 =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cov[J(Ωk1 , ωk2), J(Ωk3+r3 , ωk4+r4)

]×cov

[J(Ωk1+r1 , ωk2+r2), J(Ωk3 , ωk4)

],

and

IIIk2,k4 =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)

×cum[J(Ωk1 , ωk2), J(Ωk1+r1 , ωk2+r2), J(Ωk3 , ωk4), J(Ωk3+r3 , ωk4+r4)

].

By using (50)-(51) we obtain expressions for the Ik2,k4 , IIk2,k4 and IIIk2,k4 . We first consider

Ik2,k4 . Using (50) we have,

Ik2,k4 =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk3)cov [J(Ωk1 , ωk2), J(Ωk3 , ωk4)]

×cov[J(Ωk1+r1 , ωk2+r2), J(Ωk3+r3 , ωk4+r4)

]= Ik1,k2,M + Ik1,k2,R, (53)

where,

Ik1,k2,M =Ir1=r3

(2π)d+2T 2

T∑t1,t2,t3,t4=1

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr)

×eit1ωk2−it2ωk2+r2−it3ωk3+it4ωk4+r4dΩ,

Ik1,k2,R =1

(2π)d+2T 2

T∑t1,t2,t3,t4=1

R1,t3−t1,t4−t2eit1ωk2−it2ωk2+r2−it3ωk3+it4ωk4+r4 .

57

We first find an expression for Ik1,k2,M

Ik1,k2,M =Ir1=r3

(2π)d+2T 2

∫D|g(Ω)|2

T−1∑s1=−(T−1)

fs1(Ω)e−is1ωk4T−|s1|∑t1=1

eit1(ωk4−ωk2 )

× T−1∑s2=−(T−1)

fs2(Ω + Ωr)eis2ωk4+r2T−|s2|∑t2=1

eit2(ωk2+r2−ωk4+r4 )

dΩ

=Ir1=r3Ik2=k4Ir2=r4

(2π)d

∫D|g(Ω)|2f(Ω, ωk2)f(Ω + Ωr1 , ωk2+r2)dΩ +O

(Ir1=r3

T+ `λ,a,n

)=

Ir1=r3Ik2=k4Ir2=r4

(2π)d

∫D|g(Ω)|2f(Ω, ωk2)f(Ω + Ωr1 , ωk2+r2)dΩ +O

(Ir1=r3

T+ `λ,a,n

).

It is straightforward to show that Ik1,k2,R = O(`λ,a,n). Therefore we have

Ik1,k2 =Ir1=r3Ik2=k4Ir2=r4

(2π)d

∫D|g(Ω)|2f(Ω, ωk2)f(Ω + Ωr1 , ωk2+r2)dΩ +O

(Ir1=r3

T+ `λ,a,n

).

Using the same arguments and (51)

IIk2,k4 =1

λd

a∑k1,k3=−a

g(Ωk1)g(Ωk2)T∑

t1,...,t4=1

exp(it1ωk2 + it4ωk4+r4 − it2ωk2+r2 − it3ωk4)

cov[Jt1(Ωk1), Jt4(Ωk3+r3)

]cov

[Jt2(Ωk1+r1), Jt3(Ωk3)

]=

Ir1=r3

(2π)d+2T 2

∫Dr1

g(Ω)g(−Ω−Ωr1)

T−1∑s1=−(T−1)

fs1(Ω)eis1ωk2T−|s1|∑t1=1

eit1(ωk4+ωk2+ωr2 )

×

T−1∑s2=−(T−1)

fs2(Ω + Ωr1)e−is2ωk2+r2

T−|s2|∑t3=1

eit3(ωk2+ωr4+ωk4 )

dΩ +O(`λ,a,n)

=Ir1=r3Ik4=T−k2−r2Ir2=r4

(2π)d

∫Dr1

g(Ω)g(−Ω−Ωr1)f(Ω,−ωk2)f(−Ω−Ωr1 , ωk2+r2)dΩ

+O

(Ir1=r3

T+ `λ,a,n

). (54)

58

Using (52) (see the proof of Theorem 4.1, Jentsch and Subba Rao [2015] for details) we have

IIIk2,k4

=Ir1=r3

(2π)2dT 2

∫D2

g(Ω1)g(Ω2)T∑

t1,t2,t3,t4=1

ft2−t1,t3−t1,t4−t1(Ω1 + Ωr1 ,Ω2,−Ω2 −Ωr2)

×eit1ωk2−it2ωk2+r2−it3ωk4+it4ωk4+r4dΩ1dΩ2

+1

(2π)dT 2

T∑t1,t2,t3,t4=1

R3,t2−t1,t3−t1,t4−t1eit1ωk2−it2ωk2+r2−it3ωk4+it4ωk4+r4

=Ir1=r3

(2π)2dT 2

∫D2

g(Ω1)g(Ω2)T−1∑

s1,s2,s2=−(T−1)

fs1,s2,s3(Ω1 + Ωr1 ,Ω2,−Ω2 −Ωr1)

×eis1ωk2+r2+is2ωk4−is3ωk4+r4

T−|max(si,0)|∑t=|min(si,0)|+1

eit(ωk2−ωk2+r2−ωk4+ωk4+r4 )dΩ1dΩ2

+1

(2π)2dT 2

T−1∑s1,s2,s2=−(T−1)

R3,s1,s2,s3eis1ωk2+r2+is2ωk4−is3ωk4+r4

×T−|max(si,0)|∑t=|min(si,0)|+1

eit(ωk2−ωk2+r2−ωk4+ωk4+r4 ).

By changing the limits of the sum we have

IIIk2,k4

=Ir1=r3

(2π)2dT 2

∫D2

g(Ω1)g(Ω2)T−1∑

s1,s2,s2=−(T−1)

fs1,s2,s3(Ω1 + Ωr1 ,Ω2,−Ω2 −Ωr1)

×e−is1ωk2+r2−is2ωk4+is3ωk4+r4

T∑t=1

eit(ωk2−ωk2+r2−ωk4+ωk4+r4 )dΩ1dΩ2 +O

(`λ,a,n +

1

T 2

)=Ir1=r3Ir2=r4

(2π)2dT

∫D2

g(Ω1)g(Ω2)f(Ω1 + Ωr1 , ωk2 + ωr2 ,Ω1, ωk2 ,−Ω2 −Ωr1 ,−ωk4 − ωr2)

dΩ1dΩ2 +O

(`λ,a,nT

+Ir1=r3Ir2=r4

T 2

)(55)

59

The above results imply

λdcov[ag(ωk2 ; r1, r2), ag(ωk4 ; r3, r4)]

=Ir1=r3Ir2=r4

(2π)d

(Ik2=k4

∫Dg(Ω)g(Ω)f(Ω + Ωr1 , ωk2+r2)f(Ω, ωk2)dΩ

+Ik4=T−k2−r2

∫Dr

g(Ω)g(−Ω−Ωr1)f(Ω,−ωk2)f(−Ω−Ωr1 , ωk2+r2)dΩ

)+O

(`λ,a,n +

1

T

).

By using the well known identities

cov(<A,<B) =1

2

(<cov(A,B) + <cov(A, B)

)cov(=A,=B) =

1

2

(<cov(A,B)−<cov(A, B)

),

cov(<A,=B) =−1

2

(=cov(A,B)−=cov(A, B)

), (56)

we immediately obtain (19).

Asymptotic normality is proved using sufficient mixing assumptions.

A.2.1 PROOF of results in Section 6.1 (used in Section 4.2)

We start by analyzing the sampling properties of the first test statistic Ag,h(r1, r2).

PROOF of Lemma 6.1 We first note that

λdT

2cov

[Ag,h(r1, r2), Ag,h(r3, r4)

]= I + II + III,

where,

I =2

T

T/2∑k2,k4=1

Ik2,k4 II =2

T

T/2∑k2,k4=1

IIk2,k4 III =2

T

T/2∑k2,k4=1

IIIk2,k4

and Ik2,k4 , IIk2,k4 and IIIk2,k4 are defined in the proof of Lemma 4.2. We now obtain expres-

60

sions for these terms. By substituting the expression for Ik2,k4 in (53) into I we have

I =2Ir1=r3

(2π)d+2T

T∑t1,t2,t3,t4=1

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr)dΩ

×e−it2ωr2+it4ωr4

(1

T 2

T/2∑k2,k4=1

h(ωk2)h(ωk4)eiωk2 (t1−t2)e−iωk4 (t3−t4)

)

+2

T 3

T/2∑k2,k4=1

h(ωk2)h(ωk4)∑

t1,t2,t3,t4=1

R1,t3−t1,t4−t2eit1ωk2−it2ωk2+r2−it3ωk4 it4ωk4+r4

= IM + IR.

We first obtain a neat expression for the leading term IM . Using that the function h :

[0, π]→ R is piecewise Lipschitz continuous and the integral approximation of the Riemann

sum, we have

2

T

T/2∑k=1

h(ωk)eijωk = hj +O(T−1)

where hj = 1π

∫ π0h(ω)eijωdω and the Fourier coefficients decay at the rate |hj| ≤ C|j|−1I(j 6=

0). This approximation gives

4

(2π)2T 2

T/2∑k2,k4=1

h(ωk2)h(ωk4) exp(iωk2(t1 − t2)) exp(−iωk4(t3 − t4))

= ht1−t2ht3−t4 +O(ht1−t2T−1 + ht3−t4T

−1 + T−2).

Substituting this into IM and using that |hj| ≤ C|j|−1I(j 6= 0) gives

IM =2Ir1=r3

(2π)d+2T

T∑t1,t2,t3,t4=1

ht1−t2ht3−t4e−it2ωr2+it4ωr4

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr)dΩ

+O((log T )T−1).

By making the following change of variables, s1 = t3 − t1, s2 = t4 − t2 and s3 = t1 − t2 (so

61

t3 − t4 = s1 − s2 + s3) we have

IM =Ir1=r3

(2π)d+2T

∑s1,s2,s3,t2

hs3hs1−s2+s3eis2ωr4e−it2(ωr2−ωr4 )

∫D|g(Ω)|2fs1(Ω)fs2(Ω + Ωr1)dΩ

+O((log T )T−1)

=Ir1=r3Ir2=r4

(2π)d+2

∑s1,s2,s3

hs3hs1−s2+s3eis2ωr2

∫D|g(Ω)|2fs1(Ω)fs2(Ω + Ωr1)dΩ

+O((log T )T−1),

where in the last term we have used that T−1∑T

t=1 e−it2(ωr2−ωr4 ) = I(r1 = r2). Next we use

that∑

s3hs3hs3+(s1−s2) = 1

π

∫|h(ω)|2 exp(−iω(s1 − s2))dω to give

IM =Ir1=r3Ir2=r4

π(2π)d+2

∫ π

0

|h(ω)|2∑s1,s2

e−iω(s1−s2)eis2ωr2∫D|g(Ω)|2fs1(Ω)fs2(Ω + Ωr1)dΩdω

+O((log T )T−1)

=Ir1=r3Ir2=r4

π(2π)d

∫ π

0

∫D|h(ω)|2|g(Ω)|2f(Ω, ω)f(Ω + Ωr1 , ω + ωr2)dΩ, dω

+O((log T )T−1)

By using a similar method we can show that |IR| = O(`λ,a,n). Altogether (using that f is

real) we get

I =Ir1=r3Ir2=r4

(2π)d

∫ π

0

∫D|h(ω)|2|g(Ω)|2f(Ω, ω)f(Ω + Ωr1 , ω + ωr2)dΩdω

+O((log T )T−1 + `λ,a,n)

= Ir1=r3Ir2=r4

∫ π

0

|h(ω)|2Vg(ω; Ωr1 , ωr2)dω.

Using similar arguments we can show that

II =Ir1=r3

(2π)d+2T

T∑t1,t2,t3,t4=1

∫Dr1

g(Ω)g(−Ω−Ωr)ft4−t1(Ω)ft3−t2(Ω + Ωr)dΩ

×eit2ωr2+it4ωr4

(1

T 2

T/2∑k2,k4=1

h(ωk2)h(ωk4)eiωk2 (t1−t2)e−iωk4 (t3−t4)

)+O(`λ,a,n).

62

We set s1 = t4 − t1, s2 = t3 − t2, s3 = t1 − t2 (and t3 − t4 = s2 − s1 − s3) to give

II =Ir1=r3

(2π)d+2T

∑s1,s2,s3,t2

hs3hs2−s1−s3

∫Dr1

g(Ω)g(−Ω−Ωr)fs1(Ω)fs2(Ω + Ωr)dΩ

×eit2ωr2+i(s1+s3+t2)ωr4 +O(`λ,a,n + (log T )T−1).

By changing the limits of the sum over t2 we have

II =Ir1=r3

(2π)d+2T

∑s1,s2,s3

hs3hs2−s1−s3

∫Dr1

g(Ω)g(−Ω−Ωr)fs1(Ω)fs2(Ω + Ωr)dΩ

×ei(s1+s3)ωr4

T∑t2=1

eit2(ωr1+ωr2 )

︸ ︷︷ ︸r1=T−r2

+O(`λ,a,n + (log T )T−1) = O(`λ,a,n + (log T )T−1),

where the last line follows from the fact that r1 and r2 are constrained such that 0 ≤ r1 ≤r2 < T/2. The following expression for III follows immediately from (55).

III =Ir1=r3Ir2=r4

π2(2π)2d

∫ π

0

∫ π

0

∫D2

g(Ω1)g(Ω2)h(ω1)h(ω2)

×f4(Ω1 + Ωr1 , ω1 + ωr2 ,Ω2, ω2,−Ω2 −Ωr1 ,−ω2 − ωr2)dΩ1dΩ2dω1dω2

+O((log T )T−1 + `λ,a,n).

This gives us

λdT

2cov

[Ag,h(r1, r2), Ag,h(r3, r4)

]= Ir1=r3Ir2=r4

(1

π

∫ π

0

|h(ω)|2Vg(ω; Ωr1 , ωr2)dω +1

(2π)2dπ2

∫ π

0

∫ π

0

∫D2

g(Ω1)g(Ω2)

×h(ω1)h(ω2)f4(Ω1 + Ωr1 , ω1 + ωr2 ,Ω2, ω2,−Ω2 −Ωr1 ,−ω2 − ωr2)dΩ1dΩ2dω1dω2

)+O((log T )T−1 + `λ,a,n).

Note that 1/[(2π)dπ2] = 4/(2π)2d+2 gives the fourth order cumulant term in (41).

By using the expressions for I, II and III and (56), we obtain (41).

By using mixing-type arguments the CLT can be proved.

A.2.2 Proof of results in Section 6.1 (used in Section 4.3)

PROOF of Lemma 6.2 equation (42) Expanding cov [Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r3, r4)]

63

gives

λdHcov [Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r3, r4)] = IH + IIH + IIIH ,

where

IH =1

H

H∑k2,k4=1

Ij1H+k2,j2H+k4 ,

IIH =1

H

H∑k2,k4=1

IIj1H+k2,j2H+k4 ,

IIIH =1

H

H∑k2,k4=1

IIIj1H+k2,j2H+k4 ,

and Ik2,k4 , IIk2,k4 and IIIk2,k4 are defined in the proof of Lemma 4.2. We now find expressions

for these terms, first focusing on the case j1 = j2 = j. By using (54) we have,

IH = IH,M + IH,R,

where

IH,M =HIr1=r3

(2π)d+2T 2

T∑t1,t2,t3,t4=1

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr)dΩ× e−it2ωr2+it4ωr4

×(

1

H2

H∑k2,k4=1

h(ωjH + ωk2)h(ωjH + ωk4)ei(ωjH+ωk2 )(t1−t2)e−i(ωjH+ωk4 )(t3−t4)

)

IH,R =1

T 2H

H∑k2,k4=1

h(ωjH+k2)h(ωjH+k4)∑t1,t2,t3,t4=1

R1,t3−t1,t4−t2eit1ωjH+k2

−it2ωjH+k2+r2−it3ωjH+k4

+it4ωjH+k4+r4 .

We first bound the inner sum in IH,M . Using the approximation of the Riemann sum by an

integral we have,

1

H

H∑k=1

h(ωjH + ωk)eisωk =

T

H

∫ ω(j+1)H

ωjH

h(ω)eisωdω +O(H−1) = hs,H(ωjH) +O(H−1).

(57)

64

Applying the above to the following product gives

1

H2

H∑k2,k4=1

h(ωjH + ωk2)h(ωjH + ωk4)ei(ωjH+ωk2 )(t1−t2)e−i(ωjH+ωk4 )(t3−t4)

= ht1−t2,H(ωjH)ht3−t4,H(ωjH) +O(ht1−t2,H(ωjH)H−1 + ht3−t4,H(ωjH)H−1 +H−2).

Substituting the above into IH,M , using that

H

(2π)d+2T 2

T∑t1,t2,t3,t4=1

∫D|g(Ω)|2ft3−t1(Ω)ft4−t2(Ω + Ωr)dΩ = O(H)

and the same arguments used to bound IM in the proof of Lemma 6.1 we have,

IH,M =Ir1=r3Ir2=r4T

(2π)d+2H

∫ 2πω(j+1)H

2πωjH

∫D|h(ω)|2|g(Ω)|2f(Ω, ω)f(Ω + Ωr, ω + ωr2)dΩdω

+O(H−1 + (log T )T−1).

Using the same argument we can show that IH,R = O(`λ,a,n), which gives altogether

IH =Ir1=r3Ir2=r4T

(2π)d+2H

∫ ω(j+1)H

ωjH

∫D|h(ω)|2|g(Ω)|2f(Ω, ω)f(Ω + Ωr, ω + ωr2)dΩdω

+O(H−1 + (log T )T−1 + `λ,a,n).

Using the same methods, we can show that IIH = O(H−1 + (log T )T−1 + `λ,a,n) (since

≤ r2, r4 ≤ T/2). Finally to bound IIIH we substitute (55) into IIIH to give

IIIH =Ir1=r3Ir2=r4

(2π)2dTH

H∑k2,k4=1

∫D2

g(Ω1)g(Ω2)×

f(Ω1, ωjH+k2 ,−Ω1 −Ωr1 ,−ωjH+k2+r2 ,−Ω2,−ωjH+k4)dΩ1dΩ2

+O

(H`λ,a,nT

+HIr1=r3Ir2=r4

T 2

)By using (57) we have

IIIH =TIr1=r3Ir2=r4

H(2π)2d+2

∫ ω(j+1)H

ωjH

∫ ω(j+1)H

ωjH

∫D2

g(Ω1)g(Ω2)h(ω1)h(ω2)

×f4(Ω1 + Ωr1 ,Ω2, ω2,−Ω2 −Ωr1 ,−ω2 − ωr2)dΩ1dΩ2dω1dω2

+O((log T )T−1 + `λ,a,n +H−1).

We observe that IIIH = O(H/T ). Thus by using (56) we obtain (42) and a similar expression

65

for the imaginary parts.

PROOF of Lemma 6.2 equation (43) The proof of (43) follows immediately from (41).

Finally we consider the sampling properties of Dg,h,v;H(r1, r2).

PROOF of Lemma 6.3. To prove (44) we expand the expectation squared in terms of

covariance and expectations to give

E[λdDg,h,v;H(r1, r2)] = I + II

where

I =2H

2T

(T/2H)−1∑j=0

var[√HλdBg,h;H(ωj1H ; r1, r2)], and

II =2H

2T

(T/2H)−1∑j=0

∣∣∣E[√λdHBg,h;H(ωj1H ; r1, r2)]

∣∣∣2 .Using (42) we have

I =2H

T

T/(2H)−1∑j=0

2

2v(ωj1H)

(var[√Hλd<Bg,h;H(ωj1H ; r1, r2)]

+var[√Hλd=Bg,h;H(ωj1H ; r1, r2)]

)+O(`λ,a,n)

=2H

T

T/(2H)−1∑j=0

1

v(ωj1H)Wg,h(ωjH ; Ωr1 , ωr2) +O

(`λ,a,n +

1

H

)=

1

π

∫ π

0

Wg,h(ω; Ωr1 , ωr2)

v(ω)dω +O

(`λ,a,n +

1

H+H

T

)= Eg,h,v(Ωr1 , ωr2) +O

(`λ,a,n +

1

H+H

T

).

Next we consider the second term II. First considering the expectation we note that

E[√λdHBg,h;H(ωj1H ; r1, r2)] =

√λd/2√H

H∑k=1

h(ωjH+k)E[ag(ωjH+k; r1, r2)].

By using Lemma 4.1 we obtain bounds on E[ag(ωjH+k; r1, r2)], however, these rely on the

66

number of zeros in r1 and whether r2 is zero or not. More precisely,∣∣∣∣∣√λd/2√H

H∑k=1

h(ωjH+k)E[ag(ωjH+k; r1, r2)]

∣∣∣∣∣ = O

(λd/2H1/2

∏d−bj=1 (log λ+ log |mj|)

T Ir2−r4 6=0λd−b

).

Therefore,

II = O

(λdH[

∏d−bj=1 (log λ+ log |mj|)]2

(T Ir2−r4 6=0λd−b)2

)= o(1).

This proves (44).

To prove (45) we expand the covariance in terms of products cumulants to give

T

2Mcov

[λdDg,h,v;H(r1, r2), λdDg,h,v;H(r3, r4)

]=

2λ2dH

T

(T/2H)−1∑j1,j2=0

1

v(ωj1H)v(ωj2H)

(|cov[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)]|2

+∣∣∣cov[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)]

∣∣∣2+cum

[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)

]+E[Bg,h;H(ωj1H ; r1, r2)]cum

[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)

]+ similar terms involving the product of third and first order cumulants

).

By using that

1

λ3d

a∑k1,k3,k5=−a

g(Ωk1)g(Ωk3)g(Ωk5)cum

[Jt1(Ωk1)Jt2(Ωk1+r1), Jt3(Ωk3)Jt4(Ωk3+r3),

Jt5(Ωk5)Jt6(Ωk5+r5)

]= O

∑B3

∏(ti,tj)∈B3

ρti−tjlog3d(a)

λ2d

(58)

and

1

λ4d

a∑k1,k3,k5,k7=−a

g(Ωk1)g(Ωk3)g(Ωk5)g(Ωk7)cum

[Jt1(Ωk1)Jt2(Ωk1+r1), Jt3(Ωk3)Jt4(Ωk3+r3),

Jt5(Ωk5)Jt6(Ωk5+r1), Jt7(Ωk3)Jt8(Ωk7+r3)

]= O

∑B4

∏(ti,tj)∈B4

ρti−tjlog4d(a)

λ3d

, (59)

67

where B3 and B4 denotes the set of all pairwise indecomposable partitions of the sets

1, 2, 3×4, 5, 6 and 1, 2, 3, 4×5, 4, 6, 7 (for example, it contains the element (1, 4), (3, 6)

, (5, 8), (2, 7)) respectively, we can show that

λ3d/2cum[ag(ωk2 ; r1, r2), ag(ωk2 ; r1, r2), ag(ωk4 ; r1, r2)

]= O

(log3d(a)

λd/2

)λ2dcum

[ag(ωk2 ; r1, r2), ag(ωk2 ; r1, r2), ag(ωk4 ; r1, r2), ag(ωk4 ; r1, r2)

]= O

(log4d(a)

λd

).

From this we expect (by using the methods detailed in the proof of Lemma B.5, Eichler

[2008]), though a formal proof is not given, that the terms involving cumulants of order three

and above are asymptotically negligible. Moreover that∣∣∣cov[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)]

∣∣∣2is asymptotically negligible for j1 6= j2. Using this we have

T

2Mcov

[λdDg,h,v;H(r1, r2), λdDg,h,v;H(r3, r4)

]=

2λ2dH

T

(T/2H)−1∑j1,j2=0

1

v(ωj1H)v(ωj2H)|cov[Bg,h;H(ωj1H ; r1, r2), Bg,h;H(ωj2H ; r1, r2)]|2 .

Substituting (42) into the above gives (45).

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